lie symmetry method for partial differential equations ... · differential equations with fewer...

Università degli Studi di Trento

Dipartimento di Matematica

Corso di Laurea magistrale in Matematica

Lie symmetry methodfor partial differential equations

with application in finance

FINAL THESIS

Supervisor:Prof. Luciano TubaroCo-supervisor:Prof. Luca Di PersioCo-supervisor:Prof. Andrea Pascucci

Graduant:Francesco Giuseppe Cordoni

Accademic Year 2012-2013

i

"Per cui, quella sera,

nell’ansa che il fiume descrive costeggiando la zona degli orti per anziani,

alcuni pensionati avevano fiduciosamente messo a mollo le lenze."

Preface

Preface

Partial differential equations (PDE) arise in many areas of scientific research and they have beenclassically used in describing models spanning from quantum mechanics to biological systems. Morerecently the theory of PDE have attracted an increasing interest by mathematicians and practitionersworking in those areas related to banks and insurances.

When we study a particular differential equation, we have to face with two fundamental questions:" Is there (at least) a solution ?" and (just in case) "Is it possible to write it (them) explicitly ?"Unfortunately it does not exist a standard procedure to follow to reach these goals. Even if we areable to show that a given differential problem has solutions, we often cannot show them analytically.This makes the derivation of exact solutions for PDE a corner stone in studying differential equationstheory. In fact, although numerical analysis has made impressive strides in recent years, still the questof find analytic solution is the main goal in studying PDE. It follows that a method able to recursivelyproduce solutions of a given PDE may be very useful.

In this thesis we deeply analyse one of the most fruitful technique for finding exact solutions of agiven differential equations. Such a method, first developed by the Norwegian mathematician SophusLie (1842-1899) in the second half of the XIX century, unifies and extends ad hoc techniques in orderto produce exact solutions for PDE by exploiting the fact that many phenomena in nature has internalsymmetries. Lie developed a systematic way for finding analytic solution for a wide class of PDEsthrough continuous transformation groups. These groups are usually called Lie groups. These groupsare both a "weird" and startling merger of algebraic group, topological structures and elements ofanalysis. The core idea is to use internal symmetries admitted by a given PDE in order to reduce thenumber of independent variables. Thus a solution of the given PDE can be found solving a differentdifferential equations with fewer independent variables.

The Lie work was inspired by Galois’s theory for polynomial equations. In particular Lie tried tocreate a unified theory of integration for ordinary differential equations similar to the Abelian theorydeveloped to solve algebraic equations. For almost 100 years nobody has further studied the theory.It was in the mid 90’s that Ovsiannikov, in the ex URSS, and Bluman, in the West,have brought backthe theory to light. Since then a number of works have appeared. This is mainly due to the highalgorithmic complexity characterizing the procedure which gives the internal symmetries of a givenPDE and, thus, its solutions. A great improvement in concrete applications of the Lie approach have

iii

iv

been realized by the use of symbolic calculus and fast calculators. Modern computers can be easilyprogrammed to find the desired symmetries, making the theory even more appealing and effective.

In recent years, particularly after the worldwide financial crisis of 1987, the field of mathematicalfinance has seen a massive development leading it to became one of the most studied field in appliedmathematics. Particular attention have been attracted by problems related to the fair pricing procedurefor a huge pletora of structured financial instruments such in the case of financial derivatives. Financialderivatives are particular type of contracts whose value depends on an underlying asset. The timebehaviour of the price of such a contract is usually modeled by a partial differential equations,eventually driven by stochastic terms. In such a scenario is clearly of particular relevance theexistence of a (possibly) unique fair price. Moreover, when such a fair price is shown to exist, is ofgreat importance to explicitly write it as an analytical function of its (eventually stochastic) parameters.This is why many mathematicians have applied the Lie’s theory to obtain efficient ways to pricea wide class of financial instruments. We will see further (see Sec. [] and []) how such techniquescan be exploited to derive not only analytical solutions but also fundamental solutions and transitiondensity functions associated to particular financial models.

Since the huge realm of applications of the Lie approach, we are sadly not able to treat all possibleapplications even restricting ourselves to the financial applications. In particular we have chosen tofocus our attention on linear PDE’s problems with one single dependant variable. Particular attentionhave been given to local symmetries with emphasis on point transformations, i.e. symmetries definedby infinitesimal transformations whose infinitesimals depend on independent variable, dependentvariable and its derivatives.

We would like to underline that one of the main reason for the huge success encountered by Lietheory during recent years, relies on the fact that it provides a unified approach to treat basically.any kind of PDE. It follows that in this thesis we are able to merely scratch the surface of a widertheory. In fact symmetries more general than point symmetries, e.g. contact symmetries, Lie-

Backlund symmetries, etc., have been developed. Further non-local symmetries i.e. symmetries whoseinfinitesimals at any point x depend on the global behaviour of u(x), are largely used and perhapsthey are the area where Lie groups are actually an improvement with respect to standard approachesto PDE’s. In addition system of PDE’s depending on more than just one variable, can be successfullystudied and non-linear differential equations are the perfect application of such a method.

Outline of thesis

Actually mainly three (slightly) different approaches to the Lie groups theory do exist whichdiffer each one from another by emphasising a different characteristic of the theory. In this work weaim at both unifying different notations coming from such approaches and show how Lie theory ofgroup invariants can be applied for widely used financial models described by stochastic differentialequations.

TThe first approach is due to Ovsiannikov, see Ovsiannikov [Ovs82]. It gives a rigorous math-ematical foundation of the theory wich is based on both representation theory and transformationgroups analysis. During recent years a former student of him, Ibragimov, improved Ovsiannikov

v

work in a series of works, see e.g. Ibragimov [Ibr09], Ibragimov et al. [Ibr+94; Ibr+95], Ibragi-mov et al. [Ibr+96], and Ibragimov [Ibr94], related to both the mathematical theory of continuoustransformations and some of their application to mathematical physics.

Independetly and simultaneously to Ovsiannikov, Bluman studied Lie’s theory in the USA. Hisoriginal work, see Bluman and Cole [BC74], (together with the one made by Ovsiannikov,seeOvsiannikov [Ovs82] ) has been the first that brought back methods associated to Lie groups theoryto the attention of the mathematical community. Works by Bluman, see e.g. Bluman and Kumei[BK89], Bluman, Cheviakov, and Anco [BCA10], and Bluman and Anco [BA02], are focused onsimilarity reduction providing a highly efficient approach to find invariant solutions and mapping of awide set of differential equations

This thesis follows anyway a different approach mainly based on a geometric point of view, witha emphasis on differential geometry techniques. This approach can be found in the works of Olver,particularly in his book Olver [Olv93]. A part from the intuitive approach and examples helpingunderstand properly the theory, this book presents one of the best section exploiting the algorithm forfinding admitted symmetries. A good presentation of the topic can be found on Olver’s home pagehttp://www.math.umn.edu/~olver/sm.html.

Furthermore it has to be said that, as it was easily understandable, such a theory has an algebraicfoundation. The topic is broad and entire books are devoted to it. We do not treat any of such atheory in this thesis since it is beyond our purposes. For the interested reader we refer anyway to Hall[Hal03] and Varadarajan [Var84].

Chapter§ 1 is devoted to the theory developed by Lie. At first (almost) all preliminary notionsfundamental to the understanding of the method are stated. In section 1.1 we will introduce thefundamental tools such as Lie groups, flows, vector fields, infinitesimal generator and Lie algebras. Acomplementary part has been introduced in appendix A.1.5. The appendix is not necessary to theunderstanding of the theory but for an interested reader it better explains the mathematical foundationof the method. Section 1.2 treat the algorithm for finding admitted symmetries. It is the main reasonwhy such a method has seen such a huge development in the past years. The notion of prolongation,Jet-space and total derivative will be introduced. Then the algorithm itself is exploited. Eventuallythe main results to find any admitted symmetric group of a given PDE are stated. Section 1.3 willfocus on the concept of invariant (or self-similar) solutions. It is widely explained how to constructinvariant solutions for a PDE. The notions of invariant surface, invariant curve and invariant point

will introduced. Section 1.4 treat how to construct a mapping from a given DE to a target DE. Inparticular such a method it is largely used in finance. For instance the well known Black&Scholesequation can be mapped into the heat equation. Eventually in section 1.5 an example is provided.Everything said will be applied to the Heat equation in order to give an overview on Lie’s theory ofsymmetry groups. The main references for this chapter are Olver [Olv93], Ibragimov [Ibr09], Blumanand Kumei [BK89], and Ovsiannikov [Ovs82].

In chapter§ 2 we will focus on some papers that have been published recently. They use thesymmetric group of a certain class of PDE (in particular we will focus on the square root process)in order to retrieve fundamental solutions. The basic idea of the method is to choose a particularparameter ε such that the corresponding Lie group can be treated reduce to a Laplace transform. In

http://www.math.umn.edu/~olver/sm.html

vi

this way the transition density of a given SDE can be derived via inversion of the Laplace transform ofa fundamental solution. Section 2.1 is devoted to the statement of some general results that link SDEand parabolic PDE and to some preliminary notions on the method used by Craddock. Section 2.2will treat instead some particular case of square root process that arise in finance. Furthermore sucha section presents some necessary theorem to the derivation of fundamental and analytic solutionsas much as transition density functions. Complementary to this chapter is appendix B.1.3 wherefundamental solutions are introduced. Again a particular attention is given to the heat equationwhere its fundamental solution is found. The main references for this chapter are Craddock [Cra09],Craddock and Lennox [CL09; CL07], Craddock and Platen [CP09; CP04; CP03], Craddock andLennox [CL12], Pascucci [Pas11a], and Friedman [Fri64].

Eventually chapter §3 is devoted to the application of the results stated till then to financialmarkets. In particular we will focus on three widely used models: the The Cox-Ingersoll-Ross(CIR) model in section 3.1, the constant elasticity of variance (CEV) model in section 3.2 and thestochastic alpha-beta-rho (SABR) model in section 3.3. Pricing of zero coupon bond and optionsare shown. Since this is a mathematical thesis it is not our intention to extensively treat financial andeconomical aspects. Basic concept will be introduced to give a mild idea of what it is done but anydeeper concepts will be skipped. For a good treatment of financial mathematics we refer to Brigoand Mercurio [BM06], Cvitanic and Zapatero [CZ04], Lamberton and Lapeyre [LL08], Pascucci[Pas11a], and Shreve [Shr04]. For application to pricing we refer to Chen and Lee [CL10], Chuang,Hsu, and Lee [CHL10], Hsu, Lin, and Lee [HLL08], and Sinkala, Leach, and O’Hara [SLO08b;SLO08a]. In appendix C.1 the Longstaff model is treated.

Contents

1 Lie Group and Differential Equations 51.1 Lie Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.1.2 Basic results on manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.1.3 One-parameter transformation group . . . . . . . . . . . . . . . . . . . . . . 7

1.1.4 Lie group of transformations . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.1.5 Vector Field and Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

1.1.6 Lie brackets and Lie algebras . . . . . . . . . . . . . . . . . . . . . . . . . 22

1.2 Lie Group of Transformations for PDE’s . . . . . . . . . . . . . . . . . . . . . . . . 27

1.2.1 Invariant Subsets and Equations . . . . . . . . . . . . . . . . . . . . . . . . 27

1.2.2 Groups and Differential Equations . . . . . . . . . . . . . . . . . . . . . . . 32

1.2.3 Prolongation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

1.2.4 The Prolongation Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

1.2.5 The General Prolongation Formula . . . . . . . . . . . . . . . . . . . . . . 43

1.3 Invariance of a PDE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

1.3.1 Recall on the notion of invariants . . . . . . . . . . . . . . . . . . . . . . . 48

1.3.2 Invariance solutions and similarity reduction . . . . . . . . . . . . . . . . . 52

1.3.3 Formulation of invariace of a BVP for a PDE . . . . . . . . . . . . . . . . . 55

1.4 Construction of mapping relating differential equations . . . . . . . . . . . . . . . . 58

1.4.1 Mapping of infinitesimal generators . . . . . . . . . . . . . . . . . . . . . . 58

1.4.2 Theorems on invertible mapping . . . . . . . . . . . . . . . . . . . . . . . . 62

1.4.3 Lie’s classical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

1.5 The Heat Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

2 Fundamental solutions and transition densities 732.1 Preliminary results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

2.1.1 Group transformation of known solution . . . . . . . . . . . . . . . . . . . . 73

2.1.2 Fundamental solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

2.1.3 Integrating symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

2.1.4 Fundamental solutions as transition densities . . . . . . . . . . . . . . . . . 76

2.2 The square root process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

vii

viii CONTENTS

2.2.1 General results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 792.2.2 First Riccati equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 812.2.3 Second Riccati equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

3 Financial markets 893.1 The Cox-Ingersoll-Ross (CIR) model . . . . . . . . . . . . . . . . . . . . . . . . . 89

3.1.1 CIR transition density function . . . . . . . . . . . . . . . . . . . . . . . . . 893.1.2 Zero-Coupon Bond Price . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

3.2 The constant elasticity of variance (CEV) model . . . . . . . . . . . . . . . . . . . . 963.2.1 CEV transition density function . . . . . . . . . . . . . . . . . . . . . . . . 973.2.2 CEV option pricing model . . . . . . . . . . . . . . . . . . . . . . . . . . . 993.2.3 Mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

3.3 The Stochastic-alpha-beta-rho (SABR) model . . . . . . . . . . . . . . . . . . . . . 1073.3.1 The Heston model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

A Appendix 113A.1 Complements on Lie group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

A.1.1 Orbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113A.1.2 The differentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113A.1.3 Lie Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114A.1.4 The exponential map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116A.1.5 Infinitesimal groups actions . . . . . . . . . . . . . . . . . . . . . . . . . . 117

B Appendix 119B.1 Fundamental solutions for parabolic equations . . . . . . . . . . . . . . . . . . . . . 119

B.1.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119B.1.2 The fundamental solution of the heat equation . . . . . . . . . . . . . . . . . 120B.1.3 Fundamental solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

C Appendix 125C.1 The Longstaff Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

Introduction

Introduction

Lie’s method of group invariant has been introduced by Lie in the middle of the XIX century. Liegroups arose in connection with the problem of finding solutions of a given differential equation byquadratures also determining continuous transformation groups. Such approach provides families ofsolutions, for the assigned differential equation, by a given one via symmetry group. Classical Lietheory consider continuous transformation group which acts on the system’s graph space of dependentand independent variables of a given partial differential equation (PDE). If a differential equation isinvariant under a point symmetry, hence it is possible to find similarity solutions which are invariantunder subgroups of the whole group. A key point of the described procedure relies on the fact thatsymmetries of interest can be algorithmically determined.

Briefly a group G having the structure of an analytic manifold M such that the operationµ : (x, y) → xy−1 is analytic is called a Lie group. A Lie groups is naturally determined astransformation group and it can be identified with the smooth action of a Lie group on a manifoldsuch that any point is mapped into another point of the same manifold. Moreover local properties ofsuch transformation are sufficient to characterize the transformation globally. In the following wewill refer equivalently to Lie group or Lie group of transformations.

Lie’s idea was to solve differential equations by means of internal symmetries admitted by thegiven equation. A symmetry of a differential equation is a transformation which maps any solutionto another solution of the DE. Given a solution of a certain differential equation, knowledge of anadmitted symmetry leads to the generation of another solution. In particular given a differentialequation F = 0 and being S the set of all its solutions, then it can be shown that the collection of alltransformations mapping any solution of F = 0 into a solution of the same DE, determines a groupwhich is called the group admitted by the differential equation, or the symmetry group of the equation.We will focus in this thesis on continuous transformations.

Previous approach leads to the concept of continuous transformation groups. Multi-parametercontinuous transformation groups are composed by one- parameter groups depending on a singlecontinuous parameter. Each one-parameter group is determined by its infinitesimal transformation

of the corresponding first-order linear differential operator. In particular we will show (see Sec.1.1) the analogy between Lie group of transformations and flows of vector fields. One-parametertransformation groups and their generators are connected by means of the so-called Lie equations.

1

2 CONTENTS

The generators of multi-parameter transformation groups form specific linear spaces known as Lie

algebras. Namely, the generators of continuous groups admitted by a given differential are definedby solving an over-determined system of linear differential equations known as the determining

equations. The fundamental insight due to Lie was that the symmetry conditions for each of theone-parameter group of transformations composing a multi-parameter group of point transformationsturn out to be linear in the infinitesimals of the transformations. Being linear, these infinitesimaldetermining equations are much easier to analyse and, in a large number of cases, have been solvedexplicitly to yield the admitted group of point symmetries. The characteristic property of determiningequations is that the totality of their solutions span a Lie algebra.

Lie theory is particularly powerful since it allows to reduce the study of complex Lie groups tothe study of purely algebraic object, namely their associated Lie algebra by the Lie theorem. We havesaid how we can linearizing the transformation around the identity ε = 0 studying only its infinites-

imal generator. Thus an r−dimensional Lie group can be studied looking only at its infinitesimal

generators v1, . . . , vr. The space V generated by the infinitesimal generators (v1, . . . , vr) is theassociated Lie algebra i.e. the vector space spanned by the infinitesimal generators and it determinesthe local structure of the group. Notice that since, roughly speaking, a Lie group is a manifold, itmakes perfectly sense to talk about tangent space V around the identity of the group G.

If we are dealing with a PDE of the form

F (x, u(m)) = 0

we would like to find any one-parameter Lie group of transformations leaving invariant the surfacedetermined by the PDE. Lie theory allows us to study the admitted transformations locally, i.e. byconsidering the infinitesimal generator of the transformationx = Xε(x, u) = x+ εξ(x, u) +O(ε2)

u = Uε(x, u) = u+ εφ(x, u) +O(ε2)(1)

which continuously depends on the parameter ε. System (1) is known as the Lie point symmetry group

admitted by a PDE.

The set of all such transformations forms a continuous group called the Lie group G. Accordingto the theory developed by Lie, the construction of the symmetry group G is equivalent to thedetermination of an operator, the infinitesimal generator, of the group G having the following form

v = ξ(x, u)∂x + φ(x, u)∂u .

The set of all infinitesimal generators generates the Lie algebra. Such an algebraic structure com-pletely determines the global structure of the whole Lie group.

The group transformations corresponding to the infinitesimal transformations with the infinitesimal

CONTENTS 3

generator are obtained solving the Lie equationsdxdε = ξ(x, u), x|ε=0 = x

dudε = φ(x, u), u|ε=0 = u

. (2)

Moreover one can prove that the PDE associated to (2) is left invariant under a certain transformationif the invariance condition

pr(m)v[F (x, u(m))]∣∣∣F (x,u(m))=0

where pr(m)v denotes the prolongation of the vector field, is satisfied.The invariance condition leads to a set of linear partial differential equations called the determining

equations whose solutions give the symmetries admitted by the original PDE. The last step is whatmakes the method easily applicable in huge set od different fields, spanning from MathematicalPhysics, to Finance, etc.

Chapter 1

Lie Group and Differential Equations

1.1 Lie Groups

In this chapter we develop the fundamental theory that lies behind Lie’s theory of continuous

transformations group. In particular Lie consider all those transformations that can be represented bythe flow of an infinitesimal generator. In our work we will see that these transformations are essentiallythe flow of a vector field on manifoldM . To a given infinitesimal generator can therefore be associateda one-parameter Lie group of points transformations. Moreover from a set of infinitesimal generators

arise a multi-parameter Lie group if and only if they form a Lie algebra.

1.1.1 Introduction

We start by recalling two definitions that might look at first sight uncorrelated: the algebraicdefinition of group and the geometric definition of manifold.

They are both fundamental in the developing of Lie’s theory of continuous transformations.

Definition 1.1.1 (Group). A group is a set G together with a group operation (G, ) such that the

following hold:

(i) g h ∈ G, ∀ g, h ∈ G;

(ii) g (h k) = (g h) K, ∀ g, h, k ∈ G;

(iii) ∃ e ∈ G : e g = g e = g, ∀g ∈ G;

(iv) ∀g ∈ G ∃ g−1 ∈ G :, g g−1 = e = g−1 g.

A manifold is an object which, locally, just looks like an open subset of Euclidean space, butwhose global topology can be quite different.

Definition 1.1.2 (Manifold). An m-dimensional manifold is a set M together with a countable

collection of subsets Uα ⊂ M , called coordinate charts, and one-to-one functions χα : Uα → Vα

onto connected open subsets Vα ⊂ Rm, called local coordinate maps, which satisfies the following:

5

6 1. Lie Group and Differential Equations

(i)⋃α Uα = M ;

(ii) χβ χα−1 : χα(Uα ∩ Uβ)→ χβ(Uα ∩ Uβ) is a smooth function;

(iii) if x ∈ Uα and x ∈ Uβ are two distinct points of M , then there exist two open subsets W ⊂ Vαand W ⊂ Vβ with χα(x) ∈W and χβ(x) ∈ W such that

χ−1α (W ) ∩ χβ(W ) = ∅.

1.1.2 Basic results on manifolds

The main characteristic of manifolds is that they are coordinates free. In fact besides the basiccoordinate charts provided in the definition of a manifold, one can always adjoin many additionalcoordinate charts χ : M → V ⊂ Rm subject to the condition that, where defined, the correspondingoverlap maps χα χ−1 satisfy the same smoothness or analyticity requirements as M itself. Theprevious concept of change of coordinates is of paramount importance in differential geometry andwill be used all over the thesis. For instance, composing a given coordinate map with any localdiffeomorphism, meaning a smooth, one-to-one map defined on an open subset of Rm, will givea new set of local coordinates. Often one expands the collection of coordinate charts to includeall possible compatible charts, the resulting maximal collection defining an atlas on the manifoldM . The points in the coordinate chart Wα are identified with their local coordinate expressionsx = (x1, . . . , xm) ∈ Vα. The changes of coordinates provided by the overlap maps are then given bylocal diffeomorphisms y = η(x) defined on the overlap of the two coordinate charts.

A map F : M → N between smooth manifolds is called smooth if it is smooth in localcoordinates. In other words, given local coordinates x = (x1, . . . , xm) on M , and y = (y1, . . . , yn)

on N , the map has the form y = F (x), or, more explicitly, yi = F i(x1, . . . , xm), i = 1, . . . , n,where F = (F 1, . . . , Fn) is a C∞ map from an open subset of Rm to Rn. The definition readilyextends to analytic maps between analytic manifolds.

Definition 1.1.3. The rank of a map F : M → N at a point x ∈M is defined to be the rank of the

n×m Jacobian matrix ∂F i

∂xjof any local coordinate expression for F at the point x. The map F is

called regular if its rank is constant.

Standard transformation properties of the Jacobian matrix imply that the definition of rank isindependent of the choice of local coordinates. In particular, the set of points where the rank of F ismaximal is an open submanifold of the manifold M and the restriction of F to this subset is regular.

Theorem 1.1.4. Let F : M → N be a regular map of rank r. Then there exist local coordinates

x = (x1, . . . , xm) on M and y = (y1, . . . , yn) on N such that F takes the canonical form

y = F (x) = (x1, . . . , xr, 0, . . . , 0).

Thus, all maps of constant rank are locally equivalent and can be linearized by the introduction ofappropriate local coordinates. The places where the rank of a map decreases are singularities.

1.1 Lie Groups 7

Definition 1.1.5. A set f1, . . . , fk of smooth real-valued functions on a manifold M having

a common domain of definition is called functionally dependent if, for each x0 ∈ M , there is

neighbourhood U and a smooth function H(z1, . . . , zk), not identically zero on any subset of Rk,

such that H(f1(x), . . . , fk(x)) = 0 for all x ∈ U . The functions are called functionally independent

if they are not functionally dependent when restricted to any open subset of M.

Example 1.1.1. Let us consider the functions

f1(x, y) =x

yf2(x, y) =

xy

x2 + y2

are functionally dependent on the upper half plane y > 0 since the second can be written as afunction of the first

f2 =f1

1 + (f1)2.

For a regular family of functions, the rank tells us how many functionally independent functions itcontains.

Theorem 1.1.6. If a family of functions F is regular of rank r, then, in a neighbourhood of any point,

there exist r functionally independent functions f1, . . . , fr ∈ F with the property that any other

function f ∈ F can be expressed as

f = H(f1, . . . , fr)

Consequently, if f1, . . . , fr is a set of functions whose m×r Jacobian matrix has maximal rank rat x0, then, by continuity, they also have rank r in a neighbourhood of x0, and hence are functionallyindependent near x0.

1.1.3 One-parameter transformation group

Let M be an m-dimensional manifold. The collection τ(M) of all transformations of the set Mforms a group where the composition of mapping plays the part of the group operation. The groupτ(M) is called transformation of the set M .

Let now G be a group. The group homomorphism π : G→ τ(M) is called a representation of

the group G to τ(M). This means that the image π(g) is a transformation of the set M for everyelement g ∈ G and the following holds

π(g1) π(g2) = π(g1, g2), ∀g1, g2 ∈ G

The mappingM ×G→M ; (x, g) 7→ π(g)(x)

is called an operation of the group G on the set M .

Let T : G → τ(M) a representation of the group G to τ(M). We will consider invertible


transformation Tg in Rm given by the equation

x = T (g, x) =: Tg(x), x, x ∈ Rm (1.1)

We assume furthermore that it exist a certain value g0 of the parameter such that the transformation isthe identical transformation i.e.

x = Tg0(x)

and that is does not exist another value close to g0 such that the transformation reduces to the identicaltransformation.

In short we can say that equation (1.1) defines a single value transformation

Tg : Rm → Rm

for all g in a given interval U 3 g0. Then we have a local one-parameter group.

Definition 1.1.7 (local one-parameter transformation group). A set G of transformation Tg on a

manifold M is called one-parameter local transformation group if there exists a subinterval U0 ⊂ Ucontaining the identity e such that:

(i) ∃! e ∈ U0 such that the identity transformation I = Te ∈ G;

(ii) for any g ∈ U0, it exist g−1 ∈ U the inverse such that T−1g = Tg−1 ∈ G;

(iii) for any g1, g2 ∈ U0 it exists g3 ∈ U such that Tg1Tg2 = Tg3 ∈ G.

Definition 1.1.8 (Point transformations). A point transformation on a manifold M is a mapping

x = T (x) such that T is one-to-one and onto. The transformation corresponding to the inverse of T

is denoted by T−1.

Example 1.1.2. Let us consider the group G of all translations Tε

x = Tε(x) = x+ ε, x ∈ R, ε ∈ R

It is easy to verify that

(i)x = Tε(x) = x+ 0;

(ii)x = Tε1(x)−1 = T−ε1(x);

(iii)x = Tε1(Tε2(x)) = Tε1+ε2(x);

1.1 Lie Groups 9

Example 1.1.3. Let G be the set of the rotation around the origin Rε given by(x

y

)= Rε

(x

y

)=

(cosε −sinεsinε cosε

)(x

y

)

where ε ∈ [0, 2π[.We can again verify

(i) (x

y

)= R0

(x

y

)=

(x

y

)

(ii) (x

y

)= R−1

ε1

(x

y

)= R−ε1

(x

y

)

(iii) (x

y

)= Rε1Rε2

(x

y

)= Rε1+ε2

(x

y

)

Transformations in example 1.1.2 and example 1.1.3 are one-parameter local group of transfor-

mations. Furthermore example 1.1.3 gives a perfect example of a local group and why we requireconditions 1.1.7 to hold close to the identity. In fact in order to guarantee uniqueness of the identityelement in the rotation group we have to shrink the interval in which the parameter may vary. At thesame time we need that the composition on any two transformation still belongs to the group G. It iseasy to see that this does necessary hold for the rotation group if the two parameters are sufficientlyfrom from the identity e.

Definition 1.1.9. A group G of transformations on a manifold M is said to be continuous, if any two

transformations Tg1 , Tg2 ∈ G can be connected via a continuous set of elements within the group. In

other words, Tg1 can be continuously deformed into Tg2 within the group G.

We will in this thesis consider only continuous transformation groups. In fact Lie groups can beseen concretely as transformation groups which are continuous and analytic in the group operation.The complete understanding of this section will be useful later on the thesis since we will alwaysthink of a Lie group as a transformation acting on some manifold M .

1.1.4 Lie group of transformations

Definition 1.1.10 (r-parameter Lie group). An r-parameter Lie group is a group G which also carries

the structure of a smooth r-dimensional manifold in such a way that both the group operation and the

inversion

m : G×G→ G, m(ε, δ) = ε δ, ε, δ ∈ G

i : G→ G, i(ε) = ε−1, ε ∈ G

are smooth maps between manifolds. It is often denoted by Gr.


Therefore a Lie group is a group which is also a differentiable manifold, with the property that thegroup operation are compatible with the smooth structure. Furthermore, as we will see, Lie groupsprovide a natural framework for analysing the symmetries of differential equations.

Example 1.1.4. Let G = Rr. It is easily seen to be a manifold with a single coordinate chartU = Rr and χ = I. Let furthermore the group operation be the vector addition (x, y) 7→ x+ y. Theinversion x 7→ −x is clearly smooth. Therefore we can say that Rr is an example of an r-parameterabelian Lie group.

Example 1.1.5. Let G = SO(2) ⊂ GL2(R) the group of rotation in the plane subgroup of the2× 2 real invertible matrices. We can parametrize SO(2) as

SO(2) =

(cos ε − sin ε

sin ε cos ε

): 0 ≤ ε < 2π

We can identify G with the unit circle

S1 = (cos ε, sin ε) : 0 ≤ ε < 2π

in R2 that defines the manifold structure on SO(2).

A Lie group homomorphism is a smooth map φ : G→ H between two Lie groups which respectsthe group operations:

φ(ε δ) = φ(ε) φ(δ), ε, δ ∈ G

If φ has a smooth inverse, it determines an isomorphism between G and H . In practice we willnot distinguish between isomorphic Lie groups. For example (R+, ·) where · denotes the ordinarymultiplication is isomorphic to (R,+). The exponential function φ : R→ R+, φ(t) = et providesthe isomorphism.

We further assume through the thesis the manifold to be connected. Thus unless explicitlyotherwise stated all Lie groups are assumed to be connected. Doing so we deliberately exclude discretesymmetries such as reflections focusing n symmetries, like rotations, which can be continuouslyconnected to the identity element. Basically all the infinitesimal techniques that will be introduced inthe thesis works only on connected Lie groups.

Often we will be interested not in the full group but only in such elements that are close to theidentity. In fact Lie group is homogeneous i.e. every point looks locally like every other point. Theneighbourhood of group element a can be mapped into the neighbourhood of group element b bymultiplying a, and every element in its neighbourhood, on the left by group element ba−1 (or on theright by a−1b). This maps a into b and points near a into points near b. It is therefore necessary tostudy the neighbourhood of only one group operation in detail. Although geometrically all points areequivalent, algebraically one is special: the identity e. It is very useful and convenient to study theneighbourhood of this special group element.

Definition 1.1.11 (r-parameter local Lie group). An r-parameter local Lie group consists of connected

1.1 Lie Groups 11

open subset V0 ⊂ V ⊂ Rr containing the origin 0 and smooth maps

m : V × V → Rr

defining the group operation and

i : V0 → V

defining the group inversion such that

(associativity) x (y z) = (x y) z for x, y, z ∈ V and y z, x y ∈ V ;

(identity element) x 0 = x = 0 x ∀x ∈ V ;

(inverse) x x−1 = 0 = x−1 x ∀x ∈ V0.

These three axioms read as the normal axioms for a group with the exception that the elementsare not necessarily defined everywhere.

Example 1.1.6. We present here a non trivial example of a local one-parameter Lie group. LetV = x : |x| < 1 ⊂ R with the operation

x y =2xy − x− yxy − 1

, x, y ∈ V

It is easy to verify it is a group. The interesting part is that the inverse map

x−1 =x

2x− 1, x ∈ V

is defined for V0 = x : |x| < 1/2. Therefore we have found a local (but not global!) one-parameterLie group.

We can actually say that any local Lie group is locally isomorphic to a neighbourhood of theidentity of a global Lie group G. Since the only connected one-parameter Lie group is (R,+) we cansay that any local Lie group must coincides with some coordinate charts containing the identity ofe = 0 of (R,+). Once we know that such a global Lie group does exist we can reconstruct it justfrom the knowledge of a neighbourhood of the identity of the local Lie group.

Theorem 1.1.12. Let V0 ⊂ V ⊂ Rr be a local Lie group with group operation and inversion i.

Then there exists a global Lie group G and a coordinates charts χ : U∗ → V ∗. where U∗ contains

the identity element, such that V ∗ ⊂ V0, χ(e) = 0 and

χ(ε δ) = χ(ε) χ(δ), ε, δ ∈ U∗

and

χ(ε−1) = (χ(ε))−1, g ∈ U∗


Moreover there is a unique Lie group G∗ having the above properties. If G is any other such Lie

group, there exists a covering map π : G∗ → G which is a group homomorphism, whereby G∗ and G

are locally isomorphic Lie group.

The Previous theorem states that it exists a re-parametrization of the Lie group such that theidentity element can be take to be e = 0 and the composition law of the group to be + i.e. εδ = ε+δ.Such a parametrization is of paramount importance since it allow us to treat any transformation as aflow of a vector field, allowing us to determine the infinitesimal generator in such a way that will beseen further in the thesis.

The following gives us an explicit way to find such a parametrization.

Theorem 1.1.13. For any Lie group G it exists a re-parametrization such that the new parameter is

defined as

ε =

∫ ε

e

ds

w(s), where w(s) =

∂(s b)∂b

|b=e

In standard literature the new parameter ε is often called canonical parameter. In the proceedingwe will avoid to specify the composition law of a given one-parameter Lie group since it can be takento be the standard addition +.

Example 1.1.7. The local Lie group 1.1.6 must coincide with some coordinates charts containing0 in R.

The identity element of the local group is e = 0. Thus if we set

w(s) =∂ 2xy−x−y

xy−1

∂y|y=0 = x2 − 2x+ 1

Therefore we getε =

ε

ε− 1

Thus if we let

χ : U∗ → V ∗ ⊂ R, where χ(ε) =ε

ε− 1, t ∈ U∗ = ε < 1

we can see thatχ(ε+ δ) = χ(ε) χ(δ) =

2χ(ε)χ(δ)− χ(ε)− χ(δ)

χ(ε)χ(δ)− 1,

χ(−ε) = (χ(ε))−1 =χ(ε)

2χ(ε)− 1

We can see that such a χ satisfies the requirements of theorem 1.1.12.

Once we know that such a global Lie group does exist we can reconstruct it just from a neighbour-hood of the identity determining the local Lie group.

In practice Lie groups arise most naturally as transformations groups on some manifold M . Ingeneral a Lie group G will be realized as a group transformation of some manifold if to each groupelement ε ∈ G there is associated a map from M to itself.

1.1 Lie Groups 13

Definition 1.1.14 (Local Lie group of transformations). Let M be a manifold and G a local Lie

group. A local group of transformation acting on M is given by a local Lie group G, an open subset

U with

e ×M ⊂ U ⊂ G×M

and a smooth map Ψ : U →M such that

(i)Ψε(Ψδ(x)) = Ψεδ(x) (ε, x) ∈ U, (ε,Ψδ(x)) ∈ U, (ε δ, x) ∈ U ; (1.2)

(ii)Ψe(x) = x, ∀x ∈M and e to be the identity of the group G; (1.3)

(iii)Ψε−1(Ψε(x)) = x (ε, x) ∈ U, (ε−1,Ψε(x)) ∈ U ; (1.4)

The careful reader may now ask where actually lies the difference between one-parameter group

of transformations and one-parameter Lie group of transformations. As the name may suggest thereare similar. Roughly speaking one-parameter Lie group of transformations are one-parameter group

of transformations with in addition few properties. First of all the parameter ε is continuous i.e. itvaries in an interval. This allows us to re-parametrize the Lie group considering the identity e = 0.Further we require the transformation to be smooth w.r.t. the group operation.

In the literature it is often used the notation g x for the action of a local Lie group. In thisthesis we prefer instead the notation Ψε(x) in order to emphasize the that a Lie group can be seenas the flow of a vector field. Equivalently we will use the suggestive notation exp(εv) to stress theconnection between a Lie group and its infinitesimal generators.

In general, as already said, we will be interested in local Lie group. This implies that thetransformations exp(εv) may only be defined when the parameter ε is sufficiently close to the identitye which is taken to be 0.

Let us now give few examples to better fix ideas.

Example 1.1.8. The group of translation in Rm already introduced in example 1.1.2. Let a 6= 0

be a fixed vector in Rm and let G = R. Define then

x = Ψε(x) = x+ εa, ε ∈ R, x ∈ Rm

This is easily seen to give a global group action.

Example 1.1.9. The group SO(2) arise as the group of rotations in the plane M = R2 Rε givenby (

x

y

)= Rε

(x

y

)=

(cosε −sinεsinε cosε

)(x

y

)where ε ∈ [0, 2π[. Such a group is a local Lie group of transformations.


Example 1.1.10. The group of scaling. Let G = R+ be the multiplication group. Fix m realnumbers α1, . . . , αm non all zero. Then R+ acts on Rm by the scaling transformations

Ψλ(x) = (λα1x1, . . . , λαmxm), λ ∈ R+, x = (x1, . . . , xm) ∈ Rm

This group can be re-parametrized so that the identity element is taken to be e = 0 with law ofcomposition given by ε δ = ε + δ + εδ. Let us take in fact ε = λ − 1 in order to have thetransformation

Ψε(x) = ((1 + ε)α1x1, . . . , (1 + ε)αmxm), λ ∈ R+, x = (x1, . . . , xm) ∈ Rm

We will in the thesis focus on continuous group. Geometrically a one-parameter group G iscontinuous in the sense that any point x ∈M is carried by the group transformations into the pointsx = Ψε(x) whose locus is a continuous curve (passing through x) and is called a path curve of thegroup G. The group property means that any point of a path curve is carried by G into points of thesame curve. The locus of the images Ψε(x) is also termed the G-orbit of the point x and denoted O.See figure 1.1

Definition 1.1.15 (G-orbit). A G-orbit of a local transformation group is a minimal non empty group

invariant subset of the manifold M.

A set O ⊂M is an orbit if it satisfies:

(i) if x ∈ O, ε ∈ G and exp(εv)(x) is defined then x = exp(εv)(x) ∈ O;

(ii) if O ⊂ O and O satisfies part (i) then either O = O or O is empty.

-1.0 -0.5 0.0 0.5 1.0

-1.0

-0.5

0.0

0.5

1.0

Figure 1.1: The G-orbit of the rotation group

1.1 Lie Groups 15

1.1.5 Vector Field and Flow

This section is of paramount importance to a complete understanding of the concept of Lie group

of transformations and nevertheless the main reason why Lie groups are so widely used.

The study of Lie groups would simplify greatly if the group composition law could somehow belinearised, and this linearisation retained a substantial part of the information inherent in the originalgroup composition law. This in fact can be done. Lie algebras are constructed by linearising Liegroups. A Lie group can be linearised in the neighbourhood of any of its points, or group operations.Linearisation amounts to Taylor series expansion about the coordinates that define the group operation.What is being Taylor expanded is the group composition function. This function can be expanded inthe neighbourhoods of any group operations.

Let us suppose that C is a smooth curve on a manifold M parametrized by

φ : I →M where 0 3 I ⊂ R

At each point x = φ(ε) ∈ C the curve has a tangent vector

φ(ε) =dφ

dε=(φ1(ε), . . . , φm(ε)

).

We will then denote the tangent vector to C at x with

v|x = φ(ε) = φ1(ε)∂

∂x1+ · · ·+ φm(ε)

∂

∂xm

Example 1.1.11. The helixφ(ε) = (cos ε, sin ε)

in R3, with coordinates (x, y, z), has the tangent vector

vv|x = φ(ε) = − sin ε∂

∂x+ cos ε

∂

∂y+

∂

∂z= −y ∂

∂x+ x

∂

∂y+

∂

∂z

See figure 1.2.

Two curvesC = φ(ε), C = φ(θ)

passing through the same pointx = φ(ε∗) = φ(θ∗)

have the same tangent vector iff their derivatives agree

dφ

dε(ε∗) =

dφ

dθ(θ∗) (1.5)

It is easily seen how this concept is independent of the local coordinate system. Let x = φ(ε) =


-2

-1

0

1

2

x

-2

-1

0

1

2

y

0

1

2

3

z

Figure 1.2: Vector field and integral curves

(φ1(ε), . . . , φm(ε)) be the local coordinate expression of x = (x1, . . . , xm) and let y = ψ(x) adiffeomorphism. Then y = ψ(φ(ε)) is the local coordinate formula for the curve in terms of they-coordinates. Thus equation (1.5) holds iff

d

dεψ(φ(ε∗)) =

d

dθψ(φ(θ∗))

The tangent vector v|x = φ(ε) in x-coordinates takes the form

v|y=ψ(x) =∑j

d

dεψj(φ(ε))

∂

∂yj=∑j

∑i

∂ψj

∂xi(φ(ε))

dφk

dε

∂

∂yj(1.6)

in the y-coordinates.

The collection of all tangent vectors to all possible curve passing through a given point x ∈M iscalled the tangent space to M at x and it will be denoted by TM |x. Obviously if M is m-dimensionalso TM |x is a m−dimensional vector space spanned by ∂x1

, . . . , ∂xm provides a basis for thespace. The collection of all tangent spaces corresponding to all points x ∈M is called the tangentbundle of M denoted by TM =

⋃x∈M TM |x.

A vector field v on M is a smoothly varying assignment of tangent vectors on v|x ∈ TM |x toeach point x ∈M . In terms of local coordinates the vector field has the form

v|x = ξ1(x)∂

∂x1+ · · ·+ ξm(x)

∂

∂xm(1.7)

where each ξi(x) is an analytic function of x so that we deal exclusively with analytic vector field.The previous definition of vector field will be widely used in the thesis.

Definition 1.1.16 (Integral curve). An integral curve of a vector field v is a smooth curve x = φ(ε)

1.1 Lie Groups 17

whose tangent vector at any point coincides with the value of v at the same point

φ(ε) = v|φ(ε) , ∀ε ∈ I

In local coordinates x = φ(ε) =(φ1(ε), . . . , φm(ε)

)must be a solution of the autonomous

system of ordinary differential equationsdxidε = ξi(x), i = 1, . . . ,m

φ(0) = x0

(1.8)

Different integral curves through a given point can have different domains of definitions I and theunique one corresponding to the maximum domain of definition is called the maximal integral curve

through the point x.

Definition 1.1.17 (Flow). If v is a vector field we can denote by Ψ(x; ε) the flow generated by v the

parametrized maximal integral curve passing through x. The flow of a vector field will have the three

following properties

(i)Ψ(Ψ(x; ε); δ) = Ψ(x; δ + ε); (1.9)

(ii)Ψ(x; 0) = x; (1.10)

(iii)d

dεΨ(x; ε) = v|Ψ(x;ε) . (1.11)

Equation (1.11) simply states that v is tangent to the curve Ψ(x; ε) for a fixed x.The vector field v is called the infinitesimal generator of the action.Comparing now (1.9) and (1.10) with (1.2) and (1.3) we can see that the flow generated by a

vector field is the same as a local group action of the Lie group G = (R,+) acting on a manifold M .Thanks to theorem 1.1.12 we can thus always find a new coordinates chart such that any local Lie

group of transformations has identity element e = 0 and composition law given by ε δ = ε+ δ. Thismeans that any local Lie group can be seen as the action of a flow of a vector field and furthermorethe infinitesimal generator evaluated at x ∈M of the flow, defined as in (1.7), gives the infinitesimaltransformation of x.

Definition 1.1.18 (Infinitesimal generator). Let us consider a one-parameter Lie group of transfor-

mations exp(εv) acting on a manifold M associated to the vector field

v = ξ(x) · ∇ = ξ1(x)∂

∂x1+ · · ·+ ξm(x)

∂

∂xm(1.12)

Each ξi is called the infinitesimal of xi and the corresponding vector field v is called the infinitesimal

generator of exp(εv)


The following theorem is of paramount importance since it asserts that one-parameter local groupsare determined by their infinitesimal transformations.

Theorem 1.1.19 (Lie). Let G be a one-parameter local group of transformations. The one-parameter

family of transformations x = Ψε(x) associated with an analytic vector field v of the form (1.12) is

the unique solution to the autonomous system of ordinary differential equations (known as the Lie’s

equations) dxdε = ξ(x)

x|ε=0 = x(1.13)

where the ξi(x) are the coefficients of v at x.

System (1.13) suggests suggestive notation

exp(εv)x ≡ Ψε(x)

for a local Lie group of transformations.

Theorem 1.1.20 (Lie). Let v be a vector field. Solutions of the problem (1.13) generates the local

one-parameter Lie group for which the vector field v is its tangent vector field.

We can imagine the one-parameter group to act component wise i.e.

(xi, . . . , xm) = x = Ψε(x) =(X1ε (x), . . . , Xm

ε (x))

for some Xε smooth functions.

We will assume that one-parameter group G of transformations is given in a canonical parameter

and expand the functions Xiε into the Taylor series in a neighbourhood of the identity ε = 0. We thus

obtain the infinitesimal transformation of the group G

xi = xi + εξi(x) +O(ε2), i = 1, . . . ,m

wheredXi

ε(x)

dε

∣∣∣∣ε=0

= ξi(x),

Geometrically the infinitesimal transformation of a Lie group G defines the tangent vector

ξ(x) = (ξ1(x), . . . , ξm(x))

at the point x to the G-orbit, where the orbit of the one-parameter group action are the maximalintegral curves of the vector field v. For some deeper notion on G-orbit we refer to section A.1.1.Therefore ξ is called the tangent vector field of the group G. The tangent vector is often written inlocal coordinates as

v =∑i

ξi(x)∂

∂xi

1.1 Lie Groups 19

Theorem 1.1.21. The one-parameter group of transformations x = exp(εv)(x) is equivalent to

x = Ψε(x) = exp(εv)x = x + εvx +ε2

2v2x + . . .

=

(1 + εv +

ε2

2v2 + . . .

)x =

∞∑k=0

εk

k!vkx (1.14)

where v is defined in (1.7).

Here is how flows of vector field give rise to the transformations on the underlying space.Let exp(εv)(x) be the transformation. Then for each fixed ε, exp(εv)(x) : M → M defines atransformation on M . Different ε leads to a different transformation exp(εv) and the set of all theseis called one-parameter family of transformations. To better understand the analogy between flowand transformation let us think at the vector field as the surface velocity of a river (which would bethe manifold M in our analogy). Let us now imagine to drop a particle in such a river and let it flowsfor a given time ε. The final location of such a particle is the transformation exp(εv). In particulardifferent values of the parameter ε lead to different transformation exp(εv) and the set of all of thesetransformations give rise to a one-parameter group of transformations.

Remark 1.1.22. Uniqueness of solutions of (1.13) guarantees us that the flow generated by the

vector field v coincides with the local action of G = R on M . The transformation exp(εv) is referred

in standard literature as exponentiating the vector field v and represented by the notation

x = exp(εv)x ≡ Ψε(x)

Summarizing the connection between flows of vector fields and local one-parameter Lie group

of transformations we can say that v is the infinitesimal generator of the transformation. The orbits

of the one-parameter group action are the maximal integral curves of the vector field v. Conversely

if exp(εv)(x) is any one-parameter group of transformations acting on M then its infinitesimal

generator is obtained by specializing (1.11) at ε = 0

v|x =d

dε

∣∣∣∣ε=0

exp(εv)(x)

Furthermore there are two ways to explicitly find a one-parameter family of transformations:

(i) express the group in terms of a power series (1.14) developed from the infinitesimal generator;

(ii) solve an initial value problem (IVP). In particular we can restate (1.11) as

d

dε[exp(εv)x] = v|exp(εv)x (1.15)

We will then evaluate v|x computing (1.15) at ε = 0.


Example 1.1.12. Let M = R with coordinates x and consider the vector field

v =∂

∂x

ThenΨε(x) = exp(εv)x = exp(ε∂x)x = x+ ε

which is globally defined. For the vector field x∂x we recover the exponential

exp(εx∂x)x = eεx

since it must be the solution of the ordinary differential equationx = x;

x(0) = x;(1.16)

-1.0 -0.5 0.0 0.5 1.0

-1.0

-0.5

0.0

0.5

1.0

Figure 1.3: Vector field of the translation group

Example 1.1.13. Let us now consider the group of rotations in the plane

Ψε(x, y) = exp(εv)(x, y) = (x cos ε− y sin ε, x sin ε+ y cos ε)

Its infinitesimal generator is given byξ(x, y) = ddε

∣∣ε=0

(x cos ε− y sin ε) = −y

ν(x, y) = ddε

∣∣ε=0

(x sin ε+ y cos ε) = x(1.17)

1.1 Lie Groups 21

Thus the infinitesimal generator has the form

v = −y∂x + x∂y

It can be easily checked that it agrees with the solution of the systemdxdε = −ydydε = x

(1.18)

-1.0 -0.5 0.0 0.5 1.0

-1.0

-0.5

0.0

0.5

1.0

Figure 1.4: Vector field of the rotation group

The effect of a change of coordinates y = ψ(x) on a vector field v is determined by its effecton each individual tangent vector v|x, x ∈ M as given by (1.6). Thus if v is a vector field whoseexpression is

v =∑i

ξi(x)∂

∂xi

and y = ψ(x) is a change of coordinates, then v has the formula

v =∑j

∑i

ξi(ψ−1(y))∂ψj

∂xi

(∂

∂yj

)

in the y-coordinates.

Definition 1.1.23. A change of coordinates y = ψ(x) defines a set of canonical coordinates for the

one-parameter Lie group of transformation

x = exp(εv)(x) (1.19)


if in term of such coordinates the group becomesyi = yi, i = 1, . . . , n− 1

yn = yn + ε(1.20)

Theorem 1.1.24. For any local Lie group of transformations there exists a set of canonical coordi-

nates y = ψ(x) such that (1.19) is equivalent to (1.20).

The next result shows how by suitably choosing local coordinates we can simplify the expressionfor objects on manifold.

Proposition 1.1.25. Let us suppose v is a vector field not vanishing at x0 ∈M . Then there is a local

coordinate chart y = (y1, . . . , ym) at x0 such that in term of this coordinates v = ∂∂y1

The importance of canonical coordinates will be seen in practice in section 1.3.

Remark 1.1.26. We would like to stress the difference between canonical parameters and canonical

coordinates. They are both a change of coordinates charts. The difference is that the former is

a change of coordinates of the parameters ε such that the composition law in +. Further this is

needed if we are to consider any local Lie group as the flow of a vector field. The latter is a change

of coordinates charts on the manifold M and it is one of the basic methods for solving ordinary

differential equations with known symmetries.

As a proof of the fact let us think to the standard example of the rotation group. It is already

written in a canonical parameters ε. Let us think of example 1.1.3 where the vector field has been

found. But it is not written in canonical variables. In fact we can reparametrize the group such that

the rotation group can be seen as a translation group after a suitable change of variables. It will be

seen in 1.3.3.

1.1.6 Lie brackets and Lie algebras

The most important operator on vector fields is their Lie brackets.

Definition 1.1.27 (Lie bracket). Let us consider an r-parameter Lie groups of transformations with

infinitesimal generators vjj=1,...,r defined

vj =∑i

ξij(x)∂

∂xi(1.21)

on a manifold M . Their Lie bracket [vj , vk] is the unique vector field satisfying

[vj , vk](f) = vj (vk (f))− vk (vj (f)) (1.22)

for any smooth function f : M → R.

1.1 Lie Groups 23

In particular given two infinitesimal generators of the form

v =

m∑i1

ξi∂

∂xi, w =

m∑i=1

νi∂

∂xi

then Lie bracket of v and w is another first order operator

[v,w] =

m∑i=1

[v(νi)− w(ξi)

] ∂

∂xi=

m∑i=1

m∑j=1

[ξj∂νi

∂xj− νj ∂ξ

i

∂xj

]∂

∂xi(1.23)

Proposition 1.1.28. The Lie bracket has the following properties:

(bilinearity)[civi + cjvj , vk] = ci[vi, vk] + cj [vj , vk],

[vk, civi + cjvj ] = ci[vk, vi] + cj [vk, vj ].

with ci and cj any constants;

(skew-symmetry)[vi, vj ] = −[vj , vi];

(Jacobi identity)[vi, [vj , vl]] + [vj , [vl, vi]] + [vl, [vi, vj ]] = 0

Example 1.1.14. The Lie bracket of the scaling and translation vector fields on M = R is

[∂x, x∂x] = ∂x

reflecting the non-commutativity of the operations of translation and scaling. In particular, if the twoflows commute, then the Lie bracket of their infinitesimal generators is necessarily zero by theorem1.1.36.

Once we have introduced the Lie bracket operator we can now define the main algebraic structureof Lie group, the Lie algebra. Such a structure is of paramount importance since it is, in a certainsense, the infinitesimal generator of the Lie group G.

It will be seen in next section that the set of all solutions of the determining equations for agiven differential equation is a vector space closed with respect to the Lie bracket of infinitesimalgenerators. Such vector spaces are called Lie algebras. This section is devoted to this particular vectorspaces. They are of great importance since they are the link between one-parameter Lie group of

transformations and the infinitesimal generator.The study of a Lie group G can in fact be greatly simplified by considering the so called tangent

space V of G around the identity of the group. This tangent space V is the space generated by theinfinitesimal generator v. We have introduced an operation on this vector field so that the resultingalgebraic structure, called the Lie algebra, determines the local structure of a group. Actually the


tangent space itself is the Lie algebra. Let us notice that makes perfectly sense to talk about tangentspace since a Lie group is defined together with a manifold.

Let us consider an r-parameter Lie group of transformations. The group under consideration hasan infinite number of elements. However the properties of the group, as already mentioned, may bededuced from a finite number r of operators, called the infinitesimal generators.

Definition 1.1.29. Let G be a group acting on a manifold M . A vector field v on M is called

G-invariant if it is unchanged by the action of any group element g ∈ G.

Associated to any Lie group, there are two different Lie algebras, which we denote by gL and gR

respectively. Both Lie algebras play a role, although the right-invariant one is by far the more usefulof the two, due to the convention that Lie groups act on the left on manifolds. Therefore, in the sequel,when we talk about the Lie algebra associated with a Lie group, we shall mean the right-invariantLie algebra, and denote it by g = gR. For a mild treatment of the topic the interested reader can seeA.1.3, for a deeper treatment we refer instead to [Olv93].

Definition 1.1.30 (Lie algebra). A Lie algebra g is a vector space over some field F with an additional

law of combination of elements (the Lie bracket)

[·, ·] : g× g→ g

such that ∀a, b ∈ F and ∀vi, vj , vk ∈ g the following properties hold:

(Closure) [vi, vj ] ∈ g;

(Bilinearity) [vi, avj + bvk] = a[vi, vj ] + b[vi, vk];

(Anticommutativity) [vi, vj ] = −[vj , vi];

(Jacobi identity) [vi, [vj , vk]] + [vj , [vk, vi]][vk, [vj , vi]]=0;

A Lie algebra g spanned by r vector fields is r-dimensional. It is often denoted by gr. We will ingeneral avoid this notation if it does not create confusion.

Theorem 1.1.31. Let us consider a set of infinitesimal generators vjj=1,...,r of an r-parameter Lie

group of transformations G. Therefore the set vjj=1,...,r form an r-dimensional Lie algebra gr

Theorem 1.1.32. The Lie bracket of any two infinitesimal generators of an r-parameter Lie group of

transformations is also an infinitesimal generator, in particular

[vi, vj ] = Chijvh (1.24)

where the coefficients Chij are constants called structure constants.

1.1 Lie Groups 25

The structure of the Lie algebra is completely determined by its structure constants.

Definition 1.1.33. Equation (1.24) is called the commutation relation of the r-parameter Lie group

of transformations with infinitesimal generator (1.21).

Theorem 1.1.34. The structure constants defined in (1.24) satisfy the relations

Chij = −Chji

ChijCdhg + ChjgC

dhi + ChgiC

dhj = 0

There is a more geometric characterization of the Lie bracket of two vector field as the infinitesimalcommutator of the two one-parameter group exp(εvi) and exp(εvj).

Theorem 1.1.35. Let vi and vj be smooth vector fields on a manifold M . For each x ∈ M the

commutator

Ψε(x) = exp(−√εvj) exp(−

√εvi) exp(

√εvj) exp(

√εvi)

defines a smooth curve for sufficiently small ε ≥ 0.The Lie bracket [vi, vj ]|x is the tangent vector to

this curve at the end point Ψ0(x) = x

[vi, vj ]|x =d

dε

∣∣∣∣ε=0+

Ψε(x)

Figure 1.5: Commutator construction of Lie bracket

The previous theorem may seen at first sight useless as much as abstruse. As most of the theory itmakes sense when one thinks to the action of the Lie group of transformation as the action of a flowon a vector fields. Theorem 1.1.35 means that if we take a point x ∈M and we apply exp(

√εvi) we

end up on a point x ∈M . Applying now in the order exp(√εvj), exp(−

√εvi) and exp(−

√εvj) we


thus get a point x∗ ∈M . If we now take the Lie bracket [

vi, vj ]|x (let us remember that the Lie bracket is a vector field by definition) and we follow the integralcurve tangent to the vector [vi, vj ] and we stop "at ε" we would obtain the same point x∗ ∈ M asbefore. This can be easily seen in figure 1.5.

Theorem 1.1.36. Let vi, vj be two vector field on M . Then

exp(εvi) exp(δvj)x = exp(δvj) exp(εvi)x

for all ε, δ ∈ R, x ∈M if and only if

[vi, vj ] = 0

Definition 1.1.37. A subspace h ⊂ g is called sub-algebra of the Lie algebra g if for any vi, vj ∈ h,

[vi, vj ] ∈ h.

The most convenient way to display the structure of a give Lie algebra is the commutator table.

Definition 1.1.38 (Commutator tables). Let us consider again an r-parameter Lie groups of trans-

formations with infinitesimal generators vjj=1,...,r defined as in (1.21) on a manifold M . The

commutator table for the algebra will be the r × r table whose (i, j)−entry is [vi, vj ].

Let us notice that the table is skew-symmetric with all zeros on the main diagonal. Further thestructure constants can be easily read. In fact Chi,j is the coefficient of vh in the (i, j)−entry.

Example 1.1.15. Let us consider sl(2) the Lie algebra of special linear group SL(2) of 2 × 2

matrices with trace 0. We will take the basis

A1 =

(0 1

0 0

), A2 =

(12 0

0 − 12

), A3 =

(0 0

1 0

)

The commutator table is therefore

A1 A2 A3

A1 0 A1 −2A2

A2 −A1 0 A3

A3 2A2 −A3 0

Briefly speaking in the thesis we will mainly concern with the infinitesimal description of theaction of a Lie group 1 G on a manifold M . Let g be the Lie algebra of the Lie group G. Then forany v ∈ g the one-parameter group ε → exp εv acts on G. We can thus introduce the vector fieldv ∈ X(M) whose integral curves are of the form ε→ exp(εv)x. The first fundamental theorem of

Lie asserts that the action of G on M is described by a homomorphism of G into the group of allanalytic diffeomorphism of M . Thus we have the analogy

1we will use equivalently the terminology Lie group and Lie group of transformations.

1.2 Lie Group of Transformations for PDE’s 27

1.2 Lie Group of Transformations for PDE’s

The Lie symmetries of differential equations naturally form a group. Such a group, as we haveseen, is called Lie group and it is a point invertible transformations of both dependent and independentvariables of the PDE.

Of course in order to use the symmetry group in practice we first need to find the symmetries ofthe equation.

The first method used was to make a general change of all variables and then to force the newvariables to satisfy the original equation. This method leads to a complicated non-linear system ofdifferential equations for the transformation used. Lie proved that such approach is unnecessary. Hewas able in fact to develop an efficient method based on an infinitesimal formulation of the problem.He reduced therefore the problem to a solution of a linear system of partial differential equations. Thesolution of the determining equations will give us the symmetry transformation.

The present section is devoted to such a method developed by Lie. We will in the first sectionfocus on system of algebraic equations and then pass to the more complicated case of differentialequations.

From now on we will assume the standard multi-index notation. We will consider a PDE of theform

F (x, Dα|α|≤m) =∑|α|≤m

aα(x)Dα (1.25)

with x ∈ RN , α = (α1, . . . , αN ) a N-tupla of integers numbers, |α| = α1 + · · ·+ αN the length ofα and Dα = ∂α1

x1. . . ∂αNxN .

Anyway in the proceeding of the thesis we will refer to a general PDE of the form (1.25) as

F (x, u(m)) = 0

meaning with u(m) the dependent variable with derivatives up to order m. We chose such a notationboth to conform ourself to the existing literature and since it is the most intuitive and less heavynotation.

Further, if not specified, all definitions are as in the previous section. Moreover we will alwaysrefer to a group element ε ∈ G. From now on any manifold is consider to be N -dimensional.

1.2.1 Invariant Subsets and Equations

Let G be a group of transformations acting on the manifold M . A subset S ⊂ M is calledG-invariant if it is unchanged by the group transformations, meaning

exp(εv)(x) ∈ S whenever ε ∈ G and x ∈ S

The most important classes of invariant subsets are the varieties defined by the vanishing of one ormore functions.

An invariant of a transformation group is defined as a real-valued function whose values are


unaffected by the group transformations. The determination of a complete set of invariants of a givengroup action is a problem of supreme importance for the study of equivalence and canonical forms.In the regular case, the orbits, and hence the canonical forms, for a group action are completelycharacterized by its invariants.

Definition 1.2.1. Let G be a local one-parameter group of transformations acting on a manifold M .

A subset S ⊂M is called G−invariant and G is called symmetry group of S if whenever x ∈ S and

ε ∈ G thus exp(εv)(x) is defined and exp(εv)(x) ∈ S.

Definition 1.2.2 (Invariant function). Let G be a Lie group acting on a manifold M . An invariant of

G is a real-valued function f : M → R which satisfies for all

f(x) = f(exp(εv)(x)) = f(x), ∀ε ∈ G

Proposition 1.2.3. Let f : M → R. The following conditions are equivalent:

(i) f is a G−invariant function;

(ii) f is constant on the orbit of G;

(iii) the level sets f(x) = c of f are G−invariant subset of M .

Example 1.2.1. Let us consider the rotation group 1.1.3 acting on R2. Thus the orbits are clearlythe spheres r = |x| = constant, and hence any orthogonal invariant is a function of the radius:

f = F (r).

A fundamental problem is the determination of all the invariants of a group of transforma-tions. Note that if f1(x), . . . , fk(x) are invariants, and H(z1, . . . , zk) is any function, then f(x) =

H(f1(x), . . . , fk(x)) is also invariant. Therefore, we need only find a complete set of functionallyindependent invariants having the property that any other invariant can be written as a function ofthese fundamental invariants.

It can be proved that the number of independent local invariants of a regular transformation group

is completely determined by the orbit dimension. Indeed the following theorem holds.

Theorem 1.2.4. LetG be a Lie group acting on them−dimensional manifoldM with s−dimensional

orbits. At each x ∈ M , there exist m− s functionally independent local invariants f1, . . . , fm−s,

defined on a neighbourhood U of x, with the property that any other local invariant f defined on U

can be written as a function of the fundamental invariants:

f = H(f1, . . . , fm−s).

Moreover, in the regular case, two points y, z ∈ U lie in the same orbit of G if and only if the


invariants all have the same value

f i(y) = f i(z), i = 1, . . . ,m− s

Theorem 1.2.4 provides a complete answer to the question of local invariants of group actions.

Proposition 1.2.5. LetG be a Lie group of transformations acting on a manifoldM with infinitesimal

generator v. A function f : M → R is invariant under G if and only if

v[f ] = 0

for all x ∈M and every infinitesimal generator v ∈ g.

Thus the invariants u = f(x) of a one-parameter group with infinitesimal generator

v =N∑i=1

ξi(x)∂

∂xi

has to satisfy the first order homogeneous partial differential equation

N∑i=1

ξi(x)∂f(x)

∂xi= 0 (1.26)

The solutions of (1.26) are effectively found by the method of characteristics. We replace the partialdifferential equation by the characteristic system of ordinary differential

dx1

dξ1(x)= · · · = dxN

dξN (x)

The general solution can be written in the form

f1(x) = c1, . . . , fN−1(x) = cN−1

where the ci are constants of integration. The resulting functions f1, . . . , fN−1 form a complete setof functionally independent invariants of the one-parameter group generated by v.

We can extend the procedure to a system of algebraic equations. Let us now consider a system ofalgebraic equations of the form

Fν(x) = 0 ν = 1, . . . , l (1.27)

in which F are smooth functions defined for x on some manifold. A solution is therefore a pointx ∈M such that F (x) = 0 ∀ ν

SF = x ∈M : Fν(x) = 0, ∀ν ⊂M

Definition 1.2.6. A symmetry of a system of equations of the form (1.27) is a transformation exp(εv)

mapping any solution of the system to another solution i.e. exp(εv)(SF ) ⊂ SF .


Thus a group G is called symmetry group of the system if and only if the variety SF is a

G-invariant subset of M .

Therefore, a symmetry group of a system of equations maps solutions to other solutions: if x ∈Msatisfies (1.27) and ε ∈ G is any group element such that exp(εv)(x) is defined, then the transformedpoint x is also a solution to the system. Knowledge of a symmetry group of a system of equationsallows us to construct new solutions from old ones.

As it has already been mentioned before, the real strength of Lie group theory is that they canbe studied just looking the infinitesimal generator of the group action. This provides the key to thedetermination of the symmetry groups of a system of differential equations.

Proposition 1.2.7. Let G be a group acting on a manifold M and F : M → Rl a smooth function.

Then F is a G-invariant function iff every level set F (x) = c, c ∈ Rl is a G-invariant subset of of

M .

Theorem 1.2.8. Let G a local Lie group of transformations acting on the N -dimensional manifold

M. Let F : M → Rl with l ≤ N , define a system of algebraic equations

Fν(x) = 0 ν = 1, . . . , l

and assume the system is of maximal rank i.e. the Jacobian matrix is of rank l at any solution x. Then

G is a symmetry group of the system iff

v[Fν(x)] = 0 ν = 1, . . . , l whenever F (x) = 0 (1.28)

for every infinitesimal generator v ∈ g.

Example 1.2.2. Let us consider the group of rotation in the plane G = SO(2) with infinitesimalgenerator given by

v = −y∂x + x∂x

The unit circleS1 = x2 + y2 = 1

is an invariant subset of G as given

ζ(x, y) = x2 + y2 − 1

it easily followsv[ζ] = −2xy + 2xy = 0.

Further the maximal rank condition does hold since the gradient

∇ζ = (2x, 2y)


(a) The function F (x, y) (b) Level curves

Figure 1.6

does not vanish on S1. A less trivial example is given by the function

F (x, y) = x4 + x2y2 + y2 − 1

It is easy to see that (1.28) holds. In fact

v[F ] = −2xy(x2 + 1)−1F (x, y)

Furthermore we can show that the maximal rank condition is satisfied since(∂F

∂x,∂F

∂y

)= (4x3 + 2xy2, 2x2y + 2y)

vanishes only at (x, y) = (0, 0).

We can conclude that the solution set

(x, y) : x4 + x2y2 + y2 = 1

is a rotationally invariant subset of R2.

Note that F (x, y) is not a G-invariant function in this case. In fact most other level sets of F arenot rotationally invariant. See figure 1.6.

Therefore applying theorem 1.2.8 SO(2) is a symmetry group of F (x, y) = 0

Example 1.2.3. Consider the one-parameter group generated by the vector field

v = −y ∂∂x

+ x∂

∂y+ (1 + z2)

∂

∂z


the group transformations are

exp(εv)(x, y, z) = (x cos ε− y sin ε, x sin ε+ y cos ε,sin ε+ z cos ε

cos ε− z sin ε)

The characteristic system is thusdx

−y=dy

x=

dz

1 + z2

Solving this we can find that the functions

r =√x2 + y2 and w =

xz + y

yz + x

form a complete system of functionally independent invariants, whose common level sets describethe integral curves

1.2.2 Groups and Differential Equations

It should now be clear what we mean by symmetric group admitted by a given equation. We cannow introduce the less trivial concept of symmetry group of a differential equation.

For the sake of simplicity we will avoid the case of a system of differential equation withq dependent variables treating the simpler case of one differential equation with N independentvariables x = (x1, . . . , xN ) which we can view as local coordinates on the Euclidean space X ' RN

and a single dependent variable u coordinates on U ' R. Anyway we will refer Olver [Olv93] forthe general case of q dependent variables. The total space will be the Euclidean space E = X ×U 'RN+1 coordinatized by the independent and dependent variables. The symmetries we will focus onare diffeomorphisms on the space of independent and dependent variables:

(x, u) = exp(εv)(x, u) = (Xε(x, u), Uε(x, u)) (1.29)

for some smooth functions Xε and Uε.

These are often referred to as point transformations since they act pointwise on the total space E.However, it is convenient to specialize, on occasion, to more restrictive classes of transformations.For example, base transformations are only allowed to act on the independent variables, and so havethe form

(x, u) = exp(εv)(x, u) = (Xε(x), u)

In the case of connected groups, the action of the group can be recovered from that of its associatedinfinitesimal generators. A general vector field

v =

N∑i=1

ξi(x, u)∂xi + φ(x, u)∂u

on the space of independent and dependent variables generates a flow exp(εv), which is a local

one-parameter group of point transformations on E.


The concept of a symmetry group of a differential equation can be specified either in terms of itssolutions or its frame.

The first definition treats a symmetry group of a differential equation as a group of transformationsmapping every solution of the equation into its solution. This definition dealing with the totality ofsolutions of a given differential equation is natural, but does not give a practical method for findingall symmetries of an equation.

The second definition treats a symmetry group of a differential equation as a group of transfor-mations whose prolongation, i.e. extension to the derivatives involved in a differential equation inquestion, leaves invariant the frame of the differential equation. This differential algebraic definitiondoes not assume knowledge of solutions. Moreover, it is crucial for the infinitesimal formulation ofthe invariance of differential equations and provide a practical method for calculating symmetries bysolving so-called determining equations.

Definition 1.2.9. The symmetry group of a system of differential equations is the largest local group

of transformations acting on the independent and dependent variables of the system with the property

of mapping solutions to other solutions.

A Lie point symmetry is characterized by an infinitesimal transformation which leaves the given

differential equation invariant under the transformation of all independent and dependent variables.

We have first of all to explain how a given transformation exp(εv) in a Lie group G acts on afunction u = f(x). A solution to a differential equation will be described by a smooth functionu = f(x). The graph of the function,

Γf = (x, f(x)) : x ∈ Ω ⊂ X × U

where Ω ⊂ X is the domain of definition of f , determines a regular N−dimensional submanifold ofthe total space E = X × U .

The transform of Γf by exp(εv) is

exp(εv)(Γf ) = (x, u) = exp(εv)(x, u) : (x, u) ∈ Γf.

Notice how the set exp(εv)(Γf ) does not need to be necessarily a graph of another function u = f(x).However we can state that, since G acts smoothly and the identity leaves Γf unchanged, near theidentity the transform exp(εv)(Γf ) = Γf is the graph of some function u = f(x). We will then writef = exp(εv)(f) and call f the transform of f by exp(εv).

In general if we are to recover the transformed function under the group action f = exp(εv)(f)

we will proceed as follows.

Let us suppose the transformation is given in coordinates by

(x, u) = Ψε(x, u) = (Xε(x, u), Uε(x, u))


for some smooth functions Xε(x, u) and Uε(x, u). The graph Γf is therefore given byx = Xε(x, f(x)) = Xε(1× f)(x)

u = Uε(x, f(x)) = Uε(1× f)(x)

In order to find f explicitly we have to get rid of x in the previous system.

We know that for some ε sufficiently near the identity, the Jacobian is not singular and thus wecan invert everything obtaining

x = (Xε(1× f))−1(x)

This leads us toexp(εv)(f) = (Uε(1× f))(Xε(1× f))−1 (1.30)

This is far from being easy to compute. Let us notice anyway that in case the transformation actson the independent variables alone the previous formula is way easier.

The transformation takes in fact the form

(x, u) = exp(εv)(x, u) = (Xε(x, u), u)

Thus we can easily findu = f(x) = f(X−1

ε (x)) = f(X−ε(x))

Example 1.2.4. Consider the usual one-parameter group of rotations

exp(εv)(x, u) = (x cos ε− u sin ε, x sin ε+ u sin ε)

acting on the space E ' R2 consisting of one independent and one dependent variable. Such arotation transforms a function u = f(x) by rotating its graph. The equation for the transformedfunction f = exp(εv)(f) is given in implicit form

x = x cos ε− f(x) sin ε, u = x sin ε+ f(x) cos ε

so that u = f(x) is found by eliminating x from these two equations.

For example, if u = ax + b is affine then the point (x, u) = (x, ax + b) on the graph of f isrotated to the point

(x, u) = (x cos ε− (ax+ b) sin ε, x sin ε+ (ax+ b) cos ε)

Making x explicit

x =x+ b sin ε

cos ε− a sin ε

the transformed function is thus given by

u =sin ε+ a cos ε

cos ε− a sin ε+

b

cos ε− a sin ε(1.31)


which is defined provided cot ε 6= a, i.e., provided the graph of f has not been rotated to be vertical.

Definition 1.2.10. A function u = f(x) is said to be invariant under the transformation group G if

its graph Γf is a G-invariant subset.

Example 1.2.5. The graph of any invariant function f for the rotation group introduced inexample 1.1.3 acting on R2 must be an arc of a circle centred at the origin, so u = ±

√c2 − x2. Note

that there are no globally defined invariant functions in this case.

In general, since any invariant function’s graph must, locally, be a union of orbits, the existence ofinvariant functions passing through a point z = (x, u) ∈ E requires that the orbit O through z be ofdimension at most N , the number of independent variables, and, furthermore, be transverse i.e. itstangent space contains no vertical tangent directions. Since the tangent space to the orbit

TO|z = g|z

agrees with the space spanned by the infinitesimal generators of G, the transversality conditionrequires that, at each point, g|z contain no vertical tangent vectors.

Example 1.2.6. The infinitesimal generator of the rotation group is

v = −u∂x + x∂u

is vertical at u = 0. Thus, the rotation group fails the transversality criterion on the x−axis and thereare no smooth, rotationally invariant functions u = f(x) passing through such points.

As usual, the most convenient characterization of the invariant functions is based on an infinitesi-mal condition. Since the graph of a function is defined by the vanishing of its components u− f(x)

we have the following infinitesimal invariance condition

0 = v[u− f(x)] = φ(x, u)−N∑i=1

ξi(x, u)∂f

∂xi

which must hold whenever u = f(x), for every infinitesimal generator v ∈ g. These first orderpartial differential equations are known in the literature as the invariant surface conditions associatedwith the given transformation group.

Let us now give a more rigorous definition of a symmetry group of a differential equations.

Definition 1.2.11. Let F (x, u(m)) be a differential equation. A symmetry group of the PDE

F (x, u(m)) is a local group of transformations G acting on an open subset M of the space of

independent and dependent variables for the PDE with the property that whenever u = f(x) is a

solution of F (x, u(m)) and whenever exp(εv)(f) is defined for ε ∈ G then u = exp(εv)(f(x)) is

also a solution of the PDE.


1.2.3 Prolongation

We have now to properly define the concept of differential equations with a concrete geometricobject. In order to be able to do this we have to "prolong" the basic space X × U representing thespace of dependent and independent variables to a space that can represents all the derivatives as well.In fact since we are interested in studying the symmetries of differential equations, we need to knownot only how the group transformations act on the independent and dependent variables, but also howthey act on the derivatives of the dependent variables. In the recent years geometers have formalizedthis geometrical construction through the general definition of the jet space (or bundle) associatedwith the total space of independent and dependent variables. The jet space coordinates will representthe derivatives of the dependent variables.

We thus introduce the total space X × U (m), with U (m) := U × U1 × · · · × Um and each Unthe space of the derivatives of order n, whose coordinates represent the independent, the dependentand the derivatives of the dependent variables up to order m. The total space X × U (m) is called them− th order jet space of the underlying space X × U .

Given a smooth function u = f(x), f : X → U there is an induced function u(m) = pr(m)f(x)

called the m-th prolongation of f defined by the equation

uα = Dαf(x)

Thus pr(m)f is a function from X to the space U (m) and for each x ∈ X , pr(m)f(x) is a vectorwhose entries represent the values of f and all of its derivatives up to order m at the point x.

For instance in the case p = 2 with u = f(x, y) we have

pr(2)f(x, y) = (u;ux, uy;uxx, uxy, uyy) =

(f ;∂f

∂x,∂f

∂y;∂2f

∂x2,∂2f

∂x∂y,∂2f

∂y2

)Let us now suppose G is a local group of transformations acting on an open subset M ⊂ X × U .

There is an induced local action of G on the m-th jet space M (n) called the m-th prolongation ofG denote pr(m)G. This prolongation is defined so that it transforms the derivatives of functionsu = f(x) into the corresponding derivatives of the transformed function u = f(x). More rigorouslylet us suppose (x0, u

(m)0 ) is a given point in M (m). Chose any smooth function u = f(x) defined in

a neighbourhood of x0 whose graph lies in M and has the given derivatives at x0

u(m)0 = pr(m)f(x0)

If ε ∈ G the transformed function exp(εv)f is defined in a neighbourhood of

(x0, u0) = exp(εv)(x0, u0)

We thus determine the action of the prolonged group transformation pr(m)ε on the point (x0, u(m)0 )

by evaluating the derivatives of the transformed function exp(εv)f at x0. Explicitly

pr(m) exp(εv)(x0, u(m)0 ) = (x0, u

(m)o )


whereu

(m)0 = pr(m)(exp(εv)f)(x0).

Let nowF (x, u(m)) = 0

be a differential equations as usual in N independent variables and a single dependent variable. Weare assuming F (x, u(m)) to be a smooth map from the jet space F : X × U (m) → R

The differential equation determines a subvariety of the total jet space

SF = (x, u(m)) : F (x, u(m)) = 0

Therefore a smooth solution of the given differential equations is a smooth function u = f(x)

such thatF (x, pr(m)f(x)) = 0

whenever x lies in the domain of f . This is equivalent to the requirement that the graph

Γ(m)f = (x,pr(m)f(x)

of the m-th prolongation of f be entirely contained in the variety defined by the equation

Γ(m)f ⊂ SF

Example 1.2.7. Let N = 1 and M = R× R and consider the rotation group SO(2) consideredin example 1.2.4. Given a function u = f(x) the first prolongation is

pr(1)f(x) = (f(x), f ′(x))

For a rotation in the one-parameter group the first prolongation pr(1) exp(εv) will act on the space(x, u, p), where, in accordance with classical notation, we use p to represent the derivative coordinateux. Given a point (x0, u0, p0), we choose the linear polynomial

u = f(x) = p0(x− x0) + u0

as representative, noting that f(x0) = u0, f ′(x0) = p0. The transformed function is given by (1.31),so

f(x) =sin ε+ p0 cos ε

cos ε− p0 sin εx+

u0 − p0x0

cos ε− p0 sin ε

so thatp0 = f ′(x0) =

sin ε+ p0 cos ε

cos ε− p0 sin ε

which is defined provided p0 6= cot ε. Therefore the first prolongation of the rotation group is


explicitly given by

pr(1) exp(εv)(x, u, p) =

(x cos ε− u sin ε, x sin ε+ u cos ε,

sin ε+ p cos ε

cos ε− p sin ε

)where for convenience reason we have dropped the 0 subscript.

Definition 1.2.12. A point transformation g : M → M acting on the space M = X × U of

independent and dependent variables is called a symmetry of the system of partial differential

equations F (x, u(m)) = 0 if, whenever u = f(x) is a solution to the differential equation, and the

transformed function f = exp(εv)f is well defined, then u = f(x) is also a solution to the equation.

Theorem 1.2.13. Let M be an opened subset of X ×U and suppose F (x, u(m)) = 0 is a differential

equation defined over M with corresponding subvariety SF ⊂M (m). Suppose G is a local group of

transformation acting on M whose prolongation leaves SF invariant i.e. whenever (x, u(m)) ∈ SF

we have pr(m) exp(εv)((x, u(m))) ∈ SF ∀ε ∈ G such that this is defined. Then G is a symmetry

group of the differential equation in the sense of definition 1.2.11

As done previously with the group transformation we can define the prolongation even for thecorresponding infinitesimal generator.

Definition 1.2.14. Let M ⊂ X×U be open and suppose v is a vector field on M, with corresponding

one-parameter group exp(εv). The m-th prolongation of v, denoted by pr(m)v is defined to be

the infinitesimal generator of the corresponding prolonged one parameter group pr(m)[exp(εv)].

Therefore

pr(m)v∣∣∣(x,u(m))

=d

dε

∣∣∣∣ε=0

pr(m)[exp(εv)](x, u(m)) (1.32)

for any (x, u(m)) ∈M (m).

A vector field on M will in general take the form

pr(m)v = v∗ =

N∑i=1

ξi(x, u(m))∂

∂xi+∑|α|≤m

φα(x, u(m))∂

∂uα(1.33)

where with α we have denoted all the multi-index of orders 0 ≤ |α| ≤ m. In the case of v∗ be aprolongation pr(m)v of a vector field

v =

N∑i=1

ξi(x, u)∂

∂xi+ φ(x, u)

∂

∂u

the coefficients of v∗ must agree with the corresponding coefficients of v.

Of course the prolonged group action when restricted to just the zero-th order variables agreeswith the ordinary group action on M .

Example 1.2.8. Let us consider the rotation group SO(2) acting on X × U ' R2. The


corresponding infinitesimal generator is

v− u ∂

∂x+ x

∂

∂u

withexp(εv)(x, u) = (x cos ε− u sin ε, x sin ε+ u cos ε)

being ε the rotation angle.

We have shown in example ?? that the first prolongation takes the form

pr(1)[exp(εv)](x, u, ux) =

(x cos ε− u sin ε, x sin ε+ u cos ε,

sin ε+ ux cos ε

cos ε− ux sin ε

)The first prolongation of v is obtained by differentiating these expressions with respect to ε and settingε = 0. Thus we can easily show that

pr(1)v = −u ∂

∂x+ x

∂

∂u(1 + u2

x)∂

∂ux(1.34)

Note that the first two term in (1.34) agrees with those in v.

The following is the main theorem, due to Lie, that characterize the symmetry group G of a givenPDE.

Theorem 1.2.15 (Lie). Let us suppose

F (x, u(m)) = 0 (1.35)

is a differential equation. A group of transformation G is a symmetry group of a non degenerate

differential equation of the form (1.35) iff

pr(m)v[F (x, u(m))] = 0 whenever F (x, u(m)) = 0 (1.36)

for every infinitesimal generator v of G.

The proof follows immediately from theorem 1.2.13.

Here with non degenerate we mean a partial differential equation of maximal rank and locally

solvable as in the following definitions

Definition 1.2.16 (Maximal rank condition). Let be given a differential equation

F (x, u(m)) = 0

The system is said to be of maximal rank if the 1× (N +N (m)) Jacobian matrix

JF (x, u(m)) =

(. . . ,

∂F

∂xi, . . . ,

∂F

∂uα, . . .

)


of F with respect to the variables (x, u(m)) is of rank 1 whenever

F (x, u(m)) = 0

Definition 1.2.17 (local solvability). Let be given a differential equation

F (x, u(m)) = 0

For any point (x0, u(m)0 ) such that

F (x0, u(m)0 ) = 0

there exist a solution u = f(x) with

u(m)0 = pr(m)f(x0)

We have the following result due to Lie.

Theorem 1.2.18 (Lie). Let us consider the PDE defined on a manifold M

F (x, u(m)) = 0

The set of all infinitesimal symmetries from a Lie algebra of vector fields on M . Moreover if this Lie

algebra is finite dimensional the symmetry group of the system is a local Lie group of transformations

acting on M .

The conditions (1.36) are known as the determining equations of the symmetry group for thedifferential equation; note in particular that they are only required to hold on points (x, u(m)) ∈ SF

satisfying the equations. If we substitute the explicit formula (1.33) for the coefficients of theprolongation pr(m)v, we find that the determining equations form a large, over-determined, linearsystem of partial differential equations for the coefficients ξi and φ of v. In almost every example ofinterest, the determining equations are sufficiently elementary that they can be explicitly solved, andthereby one can determine the complete symmetry group of the equation.

Example 1.2.9. Let us consider again the rotation group G = SO(2) acting on X × U = R2.Let us consider the first order ordinary differential equation

F (x, u, ux) = (u− x)ux + u+ x = 0 (1.37)

The Jacobian matrix is thus

J =

(∂F

∂x,∂F

∂u,∂F

∂ux

)= (1− ux, 1 + ux, u− x)


which is of rank 1 everywhere. Applying the infinitesimal generator of pr(1)SO(2) we get

pr(1)v[F ] = −u∂F∂x

+ x∂F

∂u+ (1 + u2

x)∂F

∂ux=

= −u(1− ux) + x(1 + ux) + (1 + u2x)(u− x) =

= ux ((u− ux)ux + +u+ x) = uxF

(1.38)

Thus pr(1)v[F ] = 0 whenever F = 0. Thus the condition (1.36) is satisfied. We conclude that therotation group transforms solution of equation (1.37) to other solutions.

1.2.4 The Prolongation Formula

Theorem 1.2.15 is an important tool since it connects symmetry group of a differential equationwith the infinitesimal criterion of invariance of the PDE under the prolonged infinitesimal generatorof the group. We are therefore just left to find an explicit formula for the prolongation of a vectorfield.

Since the task is far from being simple we begin from some easier case in order to state the generalcase in the next section.

As stated before the inversion is not trivial but if we are in the case of a group action which actssolely on the independent variables the computation is actually a lot easier. Let us place ourselves forthe moment in this case. We are therefore considering the vector field

v =

N∑i=1

ξi(x)∂

∂xi

on the space M ⊂ X × U . The group transformations exp(εv) = exp(εv) are of the form

(x, u) = exp(εv)(x, u) = (Xε(x), u)

where xi = Xiε(x) has to satisfy

dXiε(x)

dε

∣∣∣∣ε=0

= ξi(x) (1.39)

We will denote now uj := ∂u∂xj

.

As stated in (1.30) we have

u = fε(x) = f(X−1ε (x)

)= f (X−ε(x))

Thus

uj =∂fε∂xj

(x) =

N∑k=1

∂f

∂xk(X−ε(x))

∂Xk−ε

∂xj(x)


We have that X−ε(x) = x and therefore

uj =

N∑k=1

∂Xk−ε

∂xj(Xε(x))uk

is the explicit formula for the prolonged group action on the first order derivatives.To find the infinitesimal generator of pr(1)v we have to differentiate with respect to ε and evaluate

it at the point ε = 0.

pr(1)v =

N∑i=1

ξi(x)∂

∂xi+

N∑j=1

φj(x, u(1))∂

∂uj

and from (1.32)

φj(x, u(1)) =d

dε

∣∣∣∣ε=0

N∑k=1

∂Xk−ε

∂xj(Xε(x))uk

As standard assumption we required all the functions to be smooth such that we can interchange theorder of differentiation obtaining

∂

∂xj

[dXk−εdε

](Xε(x))|ε=0 =

∂

∂xj

[dXk−εdε

∣∣∣∣ε=0

](x) =

∂ξk∂xj

(x)

where we have used (1.39) together with the fact that at ε = 0 and X0(x) = x is the identity.Furthermore we get ∑

l

∂2Xk−ε

∂xj∂xl(Xk−ε(x))

X lε

dε(x)

∣∣∣∣ε=0

= 0

It vanishes since, as already stated before, X0(x) = x is the identity. Thus at ε = 0 all second orderx-derivatives of Xε vanish.

We can now conclude that the basic prolongation formula for pr(1)v we get

φj(x, u(1)) = −N∑k=1

∂ξk∂xj

uk

As briefly mentioned previously the coefficients ξi(x) has to coincide with the coefficients of v.We now introduce the last definition before finally stating the main theorem that will allows us to

construct the symmetric group.Before giving the formal definition of total derivative we have to explain why we cannot use the

"classical" definition of derivative.Let us consider a function F depending on a set of independent variables x = (x1, . . . , xN ) and

a dependent variable u together with the set of all derivatives up to order m w.r.t. the independentvariable x (u(1), . . . , u(m)). We are interested in the derivatives of these functions with respect to allindependent variables. If we assume thus that u depends on x, we must consider all derivatives of Fwith respect to x and u(m). Basically we are treating u and its derivatives as a function of x.

Definition 1.2.19 (Total derivative). Let F (x, u(m)) be a smooth function of x, u and derivatives of

u up to order m, defined on an open subset M (m) ⊂ X × U (m). From now on we will assume that


m is the maximal order of the derivatives.

The total derivative of F with respect to xi is the unique smooth function DiF (x, um+1) defined

on M (m+1) and depending on derivatives of u up to order m+ 1 with the property that if u = f(x)

is any smooth function

DiF (x, pr(m+1)f(x)) =∂

∂xiF (x, pr(m)f(x))

Therefore DiF is obtained from P by differentiating F with respect to xi while treating all the u’s

and its derivatives as functions of x

Proposition 1.2.20. Given F (x, u(m)), the i-th total derivative of F has the general form

DiF =∂F

∂xi+∑α

uαi∂F

∂uα(1.40)

where for α = (α1, . . . , αk)

uαi =∂uα

∂xi=

∂k+1uα

∂xi∂xα1 . . . ∂xαk

In (1.40) the sum is over all the multi-indices of order |α| ≤ m where m is the highest order derivative

appearing in P

1.2.5 The General Prolongation Formula

We have finally all the necessary tools to introduce the final and key theorem.

Theorem 1.2.21. Let

v =

N∑i=1

ξi(x, u)∂

∂xi+ φ(x, u))

∂

∂u

be a vector field defined on an open subset M ⊂ X × U . The m-th prolongation of v is the vector

field

pr(m)v = v +∑α

φα(x, u(m))∂

∂uα(1.41)

defined on the corresponding space M (m) ⊂ X × U (m), the second summation being over the

multi-index α of order 0 ≤ |α| ≤ m.

The coefficient functions φα of pr(m)v are given by

φα(x, u(m)) = Dα

(φ−

N∑i=1

ξiui

)+

N∑i=1

ξiuαi (1.42)

where we have denoted by ui = ∂u∂xi

and uαi = ∂uα

∂xi

Proof. We start by proving the case n = 1.Let exp(εv) = exp(εv) the corresponding one-parameter group with transformations

(x, u) = exp(εv)(x, u) = (Xε(x, u), Uε(x, u))


Notice that ξi(x, u) = ddε

∣∣ε=0

Xiε(x, u), i = 1, . . . , p

φ(x, u) = ddε

∣∣ε=0

Uε(x, u)(1.43)

Let u = f(x) and u(1) = pr(1)f(x). Then for ε sufficiently small the transform of f by the groupelement exp(εv) is well-defined and x is given by

u = fε(x) = exp(εv)(f, x) = (Uε · (1× f)) · (Xε · (1× f))−1(x)

From the chain rule we obtain

Jfε(x) = J(Uε · (1× f))(x) · (J(Xε · (1× f))(x))−1 (1.44)

where J denote here the Jacobian matrix and x = (Xε · (1× f))(x))−1(x). Writing out the matrixentries of Jfε(x) thus provides explicit formulae for the first prolongation pr(1)v.

To find the infinitesimal generator pr(1)v we have to differentiate (1.44) with respect to ε andevaluating it at ε = 0. Recall first given an invertible matrix M(ε)

d

dε(M(ε)−1) = −M(ε)−1 dM(ε)

dεM(ε)−1

Notice furthermore that since ε = 0 corresponds to the identity transformation

X0(x, f(x)) = x, U0(x, f(x)) = f(x) (1.45)

so denoting by I the p× p identity matrix

J(X0 · (1× f))(x) = I, J(U0 · (1× f))(x) = Jf(x)

Differentiating now (1.44) and setting ε = 0 we find

d

dε

∣∣∣∣ε=0

Jfε(x) =d

dε

∣∣∣∣ε=0

J(Uε · (1× f))(x)− J(x)d

dε

∣∣∣∣ε=0

J(Xε · (1× f))(x) =

J(φ · (1× f))(x)− Jf(x)J(ξ · (1× f))(x)

Here ξ and φ are evaluated as (1.43).

Therefore the (k)− th entry is

φkα(x, pr(1)f(x)) =∂

∂xk(φα(x, f(x))−

N∑i=1

∂f

∂xi

∂

∂xk(ξi(x, f(x)))

and from the definition of total derivative

φk(x, u(1)) = Dk(φ(x, u)−N∑i=1

Dk(ξi(x, u))ui = (1.46)


Dk

(φ−

N∑i=1

ξiui

)+

N∑i=1

ξiuki

This concludes the proof for n = 1.

The general case is carried out by induction. The key is to seeM (n+1) as a subspace of (M (n))(1).

Therefore the inductive procedure for determining pr(m)v from pr(m−1)v is as follows. We regardpr(m−1)v as a vector field on M (m−1)and by first order prolongation formula can be prolonged to(M (m−1))(1). We then restrict the resulting vector field to the subspace M (m) and this will determinethe m− th prolongation pr(m)v. According to (1.46) the coefficient of ∂

∂uαkin the first prolongation

of pr(m−1)v is therefore

φkα = 1Dkφα −

N∑i=1

Dkξiuαi (1.47)

It is now sufficient to check that (1.42) solves (1.47). By induction we find

φkα = Dk

Dα

(φ−

N∑i=1

ξiui

)+

N∑i=1

ξiuαi

−

N∑i=1

Dkξiuαi =

= DkDα

(φ−

N∑i=1

ξiui

)+

N∑i=1

(Dkξ

iuαi + ξiuαik)−

N∑i=1

Dkξiuαi

= DkDα

(φ−

N∑i=1

ξiui

)+

N∑i=1

ξiuαik

Thus φkα is of the form (1.42) and the theorem is proved.

Given therefore an infinitesimal generator of the form

v =

p∑i=1

ξi(x, u)∂

∂xi+ φ(x, u)

∂

∂u

we have to solve the system of ODEdxidε = ξi(x, u), xi(0) = xi

dudε = φ(x, u), u(0) = u

(1.48)

Then the new solution will be u(x). As already said we will refer to this procedure as exponenti-ating the vector field v.

All the theorems stated in the present section lead to the formulation of Lie’s algorithm for findingsymmetries of an m-order PDE F (x, u(m)) which is assumed to be solvable and of maximal rank. Inparticular the algorithm proceed as follows:

1. given an m-order PDE F (x, u(m)) calculate pr(m)v[F (x, u(m))];

2. substitute the PDE in the expression of step 1 eliminating redundant information setting


pr(m)v[F (x, u(m))]∣∣F (x,u(m))=0

;

3. extract the determining equations from the prolongation by setting the coefficient of the deriva-tives in the dependent variables equal to 0;

4. solve the resulting determining equations.

The previous procedure will be clearer in a while when the complete method will be applied to theHeat equation.

Example 1.2.10. A general vector field on X × U ' R2 × R takes the form

v = ξ(x, t, u)∂

∂x+ τ(x, t, u)

∂

∂t+ φ(x, t, u)

∂

∂u

The first prolongation of v is

pr(1)v = v + φx∂

∂ux+ φt

∂

∂ut

where

φx =Dx(φ− ξux − τut) + ξuxxτuxt

=Dxφ− uxDxξ − utDxτ

=φx + (φu − ξx)ux − τxut − ξuu2x − τuuxut

(1.49)

φt =Dt(φ− ξux − τut) + ξuxtτutt

=Dtφ− uxDtξ − utDtτ

=φt + (φu − τt)ut − ξtux − τuu2t − ξuuxut

(1.50)

Similarly

pr(2)v = pr(1)v + φxx∂

∂uxx+ φtt

∂

∂utt+ φxt

∂

∂uxt

Example 1.2.11. Let us consider again the rotation group SO(2) with infinitesimal generator

v = −u∂x + x∂u

The first prolongation is thuspr(1)v = v + φx∂ux

whereφx = Dx(φ− ξux) + ξuxx = Dx(x+ xux)− uuxx = 1 + u2

x

as already evaluated previously in example 1.2.9.

1.3 Invariance of a PDE 47

Using formula (1.42) we further get

φxx = D2x(φ− ξux) + ξuxxx = D2

x(x+ uux)− uuxxx = 3uxuxx

Thus the infinitesimal generator of the second prolongation acting on X × U (2) is

pr(2)v = −u∂x + x∂u + (1 + u2x)∂ux + 3uuxx∂uxx

Using now theorem (1.2.15) we can deduce that the ordinary differential equation

uxx = 0

has G = SO(2) has a symmetry group since

pr(2)v[uxx] = 3uxuxx = 0 whenever uxx = 0

This is obvious since a rotation clearly takes straight lines to straight lines.

Let us consider further the function

F (x, u(2)) = uxx(1 + u2x)−

32

We can easily show thatpr(2)v[F ] = 0

for all ux and uxx. Hence by proposition 1.2.5, F is an invariant of pr(2)SO(2)

F (pr(2) exp(εv)(x, u(2))) = F (x, u(2))

for any rotation ε. But F is just the curvature of curve determined by the graph of u = f(x), s wehave just reproved the fact that the curvature of a curve is invariant under rotations.

1.3 Invariance of a PDE

We have now seen how to determine internal symmetries admitted by a given differential equationin a systematic way. In this section we will apply infinitesimal transformations to the construction ofsolutions of partial differential equations.

Invariants surface of the corresponding Lie group of point transformations leads to invariant

solutions. These solutions are obtained by solving a differential equation with fewer independentvariables than the original one.

If a one-parameter Lie group of transformations admitted by the PDE leaves the domain and theboundary condition of of a BVP invariant, then the solution of the BVP is an invariant solution andhence the given BVP is reduced to a BVP with one less independent variable.


1.3.1 Recall on the notion of invariants

We will now just recall (and actually give a better characterization) of notions of invariant definedin previous sections.

Let us now consider a one-parameter Lie group of transformations

x = exp(εv)(x) (1.51)

with infinitesimal generator

v =

N∑i=1

ξi(x, u)∂

∂xi(1.52)

Definition 1.3.1 (Invariant surface). A surface F (x) = 0 is an invariant surface for a one-parameter

Lie group of transformations (1.51) with infinitesimal generator (1.52) iff

F (x) = 0 when F (x) = 0

Definition 1.3.2 (Invariant curve). A curve F (x, y) = 0 is an invariant curve for a one-parameter

Lie group of transformations x = Xε(x, y)

y = Yε(x, y)(1.53)

with infinitesimal generator

v = ξx(x, y)∂

∂x+ ξy(x, y)

∂

∂y

iff

F (x, y) = 0 when F (x, y) = 0

Definition 1.3.3 (Invariant point). A point x is an invariant point for a one-parameter Lie group of

transformations (1.51) with infinitesimal generator (1.52) iff x ≡ x under (1.51).

Theorem 1.3.4. (i) A surface written in a solved form F (x) = xn − f(x1, . . . , xn−1) = 0 is an

invariant surface for (1.51) iff

v [F (x)] = 0 when F (x) = 0.

(ii) A curve written in a solved form F (x, y) = y − f(x) = 0 is an invariant curve for (1.53) iff

v [F (x, y)] = ξy(x, y)− ξx(x, y)f ′(x) = 0

when

F (x, y) = y − f(x) = 0

i.e. iff

ξy(x, f(x))− ξx(x, f(x))f ′(x) = 0


Theorem 1.3.4 gives a way of finding invariant surface of a given Lie group of transformations.We will now see an application of the previous theorem.

Definition 1.3.5. The family of surfaces

ω(x) = const = c

is an invariant family of surfaces for (1.51) iff

ω(x) = const = c when ω(x) = c

Definition 1.3.6. The family of curves

ω(x, y) = const = c

is an invariant family of curves for (1.53) iff

ω(x, y) = const = c when ω(x, y) = c

From these definitions it immediately follows that

c = exp(εv)c(c)

for some function exp(εv)c of c and group parameter ε.

Theorem 1.3.7. (i) A family of surfaces surface ω(x) = const = c is an invariant family of surfaces

for (1.51) iff

v [ω] = Fω(ω)

for some infinitely differentiable function Fω(ω).

(ii) A family of curves ω(x, y) = const = c is an invariant family of curves for (1.53) iff

v [ω] = ξx(x, y)∂ω

∂x+ ξy(x, y)

∂ω

∂y= Fω(ω)

for some infinitely differentiable function Fω(ω).

In order to find the invariant family of surfaces for a Lie group of transformations we canset, without loss of generality, Fω(ω) ≡ 1. This follows from the fact that if ω(x) = c is aninvariant family of surfaces then so is F (ω(x)) = F (c) for any function F . Further v [F (ω(x))] =

F ′(ω)v [ω] = F ′(ω)Fω(ω), so that setting F ′(ω) = 1Fω(ω) , we have v [F (ω)] ≡ 1.

We are now considering the PDE

F (x, u(m)) = 0 (1.54)

The notation is as before.


Definition 1.3.8. The one-parameter Lie group of transformationx = Xε(x, u)

u = Uε(x, u)(1.55)

leaves (1.54) invariant if and only of its m-th prolongation leaves the surface (1.54) invariant.

Invariance of the surface (1.54) under the m-th extension of (1.55) means that any solutionu = f(x) is mapped into some other solution u = φ(x; ε) of (1.54) under the action of the group(1.55).

Consequently the family of all solutions of PDE (1.54) is invariant under (1.55) iff (1.54) admits(1.55).

We recall the fundamental result on determining equations stated in theorems 1.2.15 and 1.2.18.

Theorem 1.3.9 (Infinitesimal criterion for invariance of a PDE). Let

v =

N∑i=1

ξi(x, u)∂

∂xi+ φ(x, u)

∂

∂u(1.56)

be the infinitesimal generator of (1.55). Let

pr(m) [v] =

N∑i=1

ξi(x, u(m))∂

∂xi+∑α

φα(x, u(m))∂

∂uα

the m-th prolongation of the infinitesimal generator of (1.56).Then (1.55) is admitted by the PDE (1.54) iff

pr(m)v[F (x, u(m))

]= 0

when

F (x, u(m)) = 0

Example 1.3.1. Let us consider the second-order PDE

uxx = g(x, u, ux, uy, uxy, uyy) (1.57)

The PDE (1.57) admits the one-parameter Lie group of translationx = Xε(x) = x

y = Yε(y) = y + ε

u = Uε(u) = u

(1.58)

since under this transformation we have

uxx = uxx, uxy = uxy, uyy = uyy, ux = ux, uy = uy


such that the surface defined by (1.57) is invariant in the (x, u, u(1), u(2))-jet space. Then for anyfunction f(x) we have

u = f(x)

is invariant under the transformation (1.58) and defines an invariant solution of (1.57) provided thatf(x) solves the second order PDE

f ′′(x) = g(x, f(x), f ′(x), 0, 0, 0)

Note now thatu = f(x, y)

defines an invariant surface of the transformation (1.58) iff

v(u− f(x, y)) = −∂f∂y

= 0 when u = f(x, y)

where v = ∂∂y is the infinitesimal generator of (1.58)

Example 1.3.2. Let us now consider the wave equation

utt = uxx (1.59)

Equation (1.59) admits the one-parameter group of scalingt = Tε(t) = εt

x = Xε(x) = εx

u = Uε(u) = u

(1.60)

sinceutt =

1

εutt, uxx =

1

εuxx

and thusutt = uxx when utt = uxx

If we change the coordinate choosing

r =t

x, s = ln(x), u

thus (1.60) becomesr = rs = s+ ln(ε)u = u

then the wave equation turn into the PDE

(1− r2)urr − uss + 2rurs + us − 2rur = 0 (1.61)


Correspondingly

u = f(r) = f(t

x) (1.62)

defines an invariant solution of (1.61) and hence of the wave equation (1.59) provided that f(r)

satisfies the ODE(1− r2)f ′′(r)− 2rf ′(r) = 0

The infinitesimal generator of the transformation is given by

v = t∂

∂t+ x

∂

∂x

Therefore u = f(t, x) defines an invariant surface of the transformation (1.60) iff

v[u− f(t, x)] = 0 when u = f(t, x)

ifft∂f

∂t+ x

∂f

∂x= 0

The corresponding characteristic equations

dt

t=dx

x=df

0

lead to the invariant form (1.62).

1.3.2 Invariance solutions and similarity reduction

We have since now just recalled notions already introduced previously. We are now to discussthe invariance condition of a partial differential equation to reduce the equation to an ODE or to aPDE with less independent variables. Such procedure has many idea in common (it is actually a"generalization") with a branch of analysis know as dimensional analysis. For a mild treatment of thetopic we refer to the first chapter of Bluman and Kumei [BK89].

The reduction procedure generates a similarity representation (a physical interpretation of asimilarity solution is a flow which "looks the same" either at all times, or at all length scales) of theoriginal equation. Thus a similarity reduction results into a representation which has some advantagecompared with the original equation.

Let us consider again a PDE of the from (1.54) wich admits a one-parameter Lie group oftransformations with infinitesimal generator (1.56).

Definition 1.3.10. A solution u = f(x) is an invariant solution of (1.54) corresponding to (1.56)admitted by (1.54) iff

(i) u = f(x) is an invariant surface of (1.56);

(ii) u = f(x) solves (1.54).

Therefore we can say that u = f(x) is an invariant solution of (1.54) iff u satisfies


(i) v [u− f(x)] = 0 when u = f(x) i.e

ξi(x, u)∂u

∂xi= φ(x, u); (1.63)

(ii) F (x, u(m)) = 0.

Equation (1.63) is called invariant surface condition for invariant solutions.

Invariants solutions can be determined in two ways:

Invariant Form Method

Theorem 1.3.11. A one-parameter local group G of transformations in RN has precisely

N − 1 functionally independent invariants. Any set of independent invariants

(ψ1(x) = C1, . . . , ψN−1(x) = CN−1)

determines a basis of invariants of G. The basis of course is not unique.

An arbitrary invariant F (x) of G is given by the formula

F = Φ(ψ1(x), . . . , ψN−1(x))

With this method we solve the first order PDE (1.63) by solving the corresponding characteristicequation for u = f(x)

dx1

dξ1(x, u)=

dx2

dξ2(x, u)= · · · = dxN

dξN (x, u)=

du

dφ(x, u)(1.64)

If(ψ1(x, u), . . . , ψN−1(x, u)), F (x, u)

are N independents invariants of (1.64) then the solution u = f(x) is given implicitly by theinvariant form

F (x, u) = Φ(ψ1(x, u), . . . , ψN−1(x, u)) (1.65)

where Φ is an arbitrary function of (ψ1(x, u), . . . , ψN−1(x, u)).

This reduced order PDE is obtained by substituting (1.65) into (1.54). It is straightforward thatif N = 2 then the reduced PDE is an ODE.

Theorem 1.3.12. Let us now consider the change of coordinates (x′1, . . . , x′N ) for the group

G of transformations defined by

x′1 = φ(x), x′2 = ψ1(x) . . . , x′N = ψN−1(x)

where φ is a solution of

v[φ] = 1


and (ψ1(x), . . . , ψN (x)) is a basis of invariants of the group G. Then (x′1, . . . , x′N ) is a set of

canonical coordinates.

Recall that for canonical variables exist for any one-parameter group. It is a change ofcoordinates such that under the new coordinate system the transformation is the translationgroup.

Example 1.3.3. Let us consider the rotation group G = SO(2) defined byx = x cos ε− y sin ε

y = x sin ε+ y cos ε(1.66)

with infinitesimal generatorv = −y∂x + x∂y

Solving the systemdx

−y=dy

x

leads to the invariantρ =

√x2 + y2

withv[ρ] = 2xy − 2xy = 0

We now know that any other invariants of the rotation group is of the form θ(y) = Φ(ρ). Fromtheorem 1.3.12 we require

v[θ] = 1

Thus we get

xdθ

dy= 1 ⇒ dθ

dy=

1√ρ2 − y2

θ = arctan

(y√

ρ2 − y2

)

Recalling the formula

arctan(x) = arcsin

(x√

1 + x2

)we get

θ = arcsin

(y

ρ

)Thus the canonical coordinates for the rotation group are the polar coordinates

(ρ, θ) = (√x2 + y2, arcsin

(y

ρ

))


Explication now fo (x(θ, ρ), y(θ, ρ)) leads tox = ρ cos θ

y = ρ sin θ(1.67)

and we can easily seen that the transformation is expressed as a translation in the formρ = ρ

θ = θ + ε(1.68)

with infinitesimal generatorv = ∂θ

Direct Substitution Method This method in used when it is impossible to solve the invariant surfacecondition (1.63).

Without loss of generality we assume ξn(x, u) 6= 0. Then (1.63) becomes

un = −n−1∑i=1

ξi(x, u)

ξn(x, u)ui +

φ(x, u)

ξn(x, u)(1.69)

Therefore after substituting (1.69) into (1.54) for all term involving derivatives of u with respectto xn we obtain a reduced order PDE with dependent variable u and independent variables(x1, . . . , xn−1) and parameter xn.

1.3.3 Formulation of invariace of a BVP for a PDE

The application of infinitesimal transformations to BVP’s for PDE’s is much more restrictive thanfor the ODE’s. In fact if for an ODE any BVP is automatically reduced to a BVP for the reducedorder ODE, it not usually hold for PDE’s as well. In the case of a PDE an invariant solution arisingfrom an admitted infinitesimal generator solves a given BVP if the infinitesimal generator leaves allboundary conditions invariant. If the PDE is linear anyway the situation is a little bit less restrictive.

Let us as usual consider a PDE of the form

F (x, u(m)) = 0 (1.70)

defined on a domain Ω with boundary conditions

BJ(x, u(m−1)) = 0 (1.71)

prescribed on the boundary surfaces

ωJ = 0, J = 1, . . . , s (1.72)


Let us further assume that the BVP has a unique solution. Consider now an infinitesimal generator ofthe form

v =

N∑i=1

ξi(x, u)∂

∂xi+ φ(x, u)

∂

∂u(1.73)

which defines a one-parameter Lie group of transformations in x-space as well in (x,u)-space.

Definition 1.3.13. The infinitesimal generator v is admitted by the BVP iff

(i)pr(m)v

[F (x, u(m))

]= 0 when F (x, u(m)) = 0; (1.74)

(ii)v [ωJ(x)] = 0 when ωJ(x) = 0, J = 1, . . . , s; (1.75)

(iii)

pr(m−1)v[BJ(x, u(m−1))

]= 0 when BJ(x, u(m−1)) = 0 on ωJ(x) = 0, J = 1, . . . , s

(1.76)

Theorem 1.3.14. Let the BVP admits the infinitesimal generator (1.73). Let

(ψ1(x, u), . . . , ψN−1(x, u))

be N-1 independent group invariants of (1.73) depending only on x. Let F (x, u) be a group invariant

of (1.73). Then the BVP reduces to

G(X,F (m)) = 0

defined on some domain ΩX in X-space with boundary conditions

CJ(X,F (m−1)) = 0

prescribed on boundary surfaces

νJ(X) = 0

Condition (1.75) means that each boundary surface ωJ(x) = 0 is an invariant surface νJ(X) = 0

of the infinitesimal generator

ξi(x)∂

∂xi

which is the restriction of X to the x-space. The solution of (1.70) together with the BVP is aninvariant solution

F = Φ(ψ1(x, u), . . . , ψN−1(x, u))

The invariant solution u = f(x) defined by (1.70) satisfies

v [u− f(x)] = 0 when u = f(x)


i.e.ξi(x)

∂u

∂xi= φ(x, u)

Theorem 1.3.15. If the infinitesimal generator v is of the form

v =

N∑i=1

ξi(x)∂

∂xi+ h(x)u

∂

∂u

then F is of the form

F =u

g(x)

for a known function g(x) and hence an invariant solution arising from v is of the separated from

u = g(x)Φ(X)

for an arbitrary function Φ(X) of

(ψ1(x, u), . . . , ψN−1(x, u))

If instead the PDE (1.70), again together with the BVP, admits an r-parameter Lie group oftransformations with infinitesimal generators of the form

vi =∑j

ξij(x)∂

∂xj+ φi(x, u)

∂

∂u, i = 1, . . . , r (1.77)

the unique solution u = f(x) is an invariant solution satisfying

vi(u− f(x)) = 0 i = 1, . . . , r

Theorem 1.3.16. Suppose the BVP admits an r-parameter Lie group of transformations with in-

finitesimal generator of the form

vi =∑j

ξij(x)∂

∂xj+ hi(x)u

∂

∂u, i = 1, . . . , r (1.78)

Let R be the rank of the r × n matrix

T =

ξ11(x) · · · ξ1n

.... . .

...

ξr1(x) · · · ξrn(x)

(1.79)

Let q = n−R and let Y1(x), . . . ,Yq(x) be a complete set of functionally independent invariants of

(1.78) satisfying

ξij(x)∂Yl(x)

∂xj= 0, i = 1, . . . , r, l = 1, . . . , q


Let

F =u

g(x)(1.80)

with g a known function be an invariant of (1.78) satifying

vi [w] = 0, i = 1, . . . , r

Then the BVP reduces to a BVP with q = N − r independent variables Y = (Y1, . . . ,Yq) and

dependent variable (1.80). The solution of the BVP is an invariant solution of separated form

u = g(x)Φ(Y)

where the function Φ(Y) is to be determined.

1.4 Construction of mapping relating differential equations

In section 1.3 we considered the construction of the solution of a given PDE F (x, u(m)) = 0

from other known solution of the PDE or as invariant solution.

In the present section we exploit a different approach. We will in fact use the infinitesimaltransformation, and in particular the Lie algebra, to construct a transformation which maps a givenPDE into another differential equation, the target DE, in the sense that any solution of the given DEis mapped into a solution of the target DE.

We will treat in particular only one-to-one mapping. This means that the map must establish aone-to-one correspondence between infinitesimal generators of the given and the target PDE. To bemore precise it is necessary that any Lie algebra of the infinitesimal generators of the given PDE to beisomorphic to a lie algebra of the infinitesimal generators of the target PDE. In particular such a mapis invertible. Anyway non invertible mapping can be constructed as well. Notation, if not otherwisespecified, is as in the previous sections.

This procedure will be use in section 3.2 and in appendix C.1.

For a deeper treatment of the topic we refer to both [BK89; BCA10].

1.4.1 Mapping of infinitesimal generators

Let us consider a PDE of the form

F (x, u(m)) = 0 (1.81)

with N independent variables x = (x1, . . . , xN ) and a single dependent variable u. As usual thewhole procedure can be extended to the general case of a system of DE’s with N dependent variablesand q dependent variables. For such a problem we refer to [BK89; BCA10].

We will use the following notation:

Gx : the group of all admitted continuous transformation;

1.4 Construction of mapping relating differential equations 59

gx : the Lie algebra of Gx;

Hx : a subgroup of continuous transformations Hx ⊂ Gx;

hx : the Lie algebra of Hx, hx ⊂ gx;

exp(εv)(x) : a one-parameter transformations exp(εv)(x) ∈ Hx;

v : the infinitesimal generator of exp(εv)(x), v ∈ hx of the form

v =∑i

ξxi ∂xi + νx∂u.

We consider the target PDEP (y, v(m)) = 0 (1.82)

with again N independent variables y = (y1, . . . , yN ) and a single dependent variable v. We will inparticular use the following notation:

Gy : the group of all admitted continuous transformation;

gy : the Lie algebra of Gy;

Hy : a subgroup of continuous transformations Hy ⊂ Gy;

hy : the Lie algebra of Hy , hy ⊂ gy;

exp(εv)(y) : a one-parameter transformations exp(εv)(y) ∈ Hy;

w : the infinitesimal generator of exp(εv)(y), w ∈ hy of the form

v =∑i

ξyi ∂xi + νy∂u.

Let µ denote a mapping which transforms any solution u = U(x) of equation (1.81) into asolution v = V (y) of equation (1.82). Of course this transformation need not to exist a priori and


even if it does it is unknown. Our goal is to establish a method that allows us to retrieve such atransformation. We must in particular restrict µ to a specific mapping of the formy = φ(x, u, . . . , u(l))

v = ψ(x, u, . . . , u(l))(1.83)

We will denote by Ml the class of mapping of the form (1.83) which depends on at most the l-thderivatives of u. In particular any map µ ∈Mm induces an action as in following diagram

F (x, u(m))exp(εv)(x)−−−−−−→ F (x, u(m))

µ

y yµP (y, v(m)) −−−−−−→

exp(εv)(y)P (y, v(m))

(1.84)

The relationship between Gx, Hx, Gy , Hy and µ is illustrated in figure 1.7.

The mapping µ must take any symmetry exp(εv)(x) into a symmetry exp(εv)(y)(y) such that thecomposition transformations

µ exp(εv)(x)(x) and exp(εv)(y)(y) µ (1.85)

yield the same action on the (x, u)− space.

Specifically

µ exp(εv)(x) = µ(x, u) =(φ(x, u, . . . , u(m)), ψ(x, u, . . . , u(m))

)(1.86)

with x = x+ εξx(x, u, . . . , u(m)) +O(ε2),

u = u+ ενx(x, u, . . . , u(m)) +O(ε2),

u(j) = u(j) + ενx(j)(x, u, . . . , u(m+j)) +O(ε2), j = 1, . . . , l

(1.87)

On the other hand

exp(εv)(y)(y) µ(x, u) = exp(εv)(y)(y)(y, v) = (y, v) =(φ(x, u, . . . , u(m)), ψ(x, u, . . . , u(m))

)(1.88)

with y = y + εξy(y, v, . . . , v(m)) +O(ε2),

v = v + ενy(y, v, . . . , v(m)) +O(ε2)(1.89)

such that ξy(y, v, . . . , v(m)) = ξy(φ, ψ, . . . , ψ(m)),

νy(y, v, . . . , v(m)) = νy(φ, ψ, . . . , ψ(m)),(1.90)

1.4 Construction of mapping relating differential equations 61

with φ = φ(x, u, . . . , u(l))

ψ = ψ(x, u, . . . , u(l))

ψ(j) = ψ(j)(x, u, . . . , u(l+j)) j = 1, . . . ,m

(1.91)

Equating now the O(ε2) terms in equation (1.86) and (1.88) we have, written in compact form, thenecessary conditions that the mapping µ has to satisfyv(m)φ = wy,

v(m)ψ = wv(1.92)

Figure 1.7: The mapping µ

Conditions (1.92) play the essential role in all the algorithm we develop in the present section.In fact in order to such a transformation from the given PDE to the target PDE to exists, it isnecessary that each infinitesimal generator v ∈ hx corresponds to some infinitesimal generatorw ∈ hy. Moreover the components (φ, ψ) of the transformation µ must satisfy conditions (1.92).These conditions play in a sense the same role the determining equations play in the determination ofthe admitted symmetries of a given PDE. Basically they reduce the nature of the dependence of themapping µ. In particular without such a conditions it is often impossible (and anyway particularlyboring) to determine whether or not there exist a mapping µ ∈Ml.


1.4.2 Theorems on invertible mapping

Theorem 1.4.1. Lei Gi be Lie groups and gi the corresponding Lie algebras, i = 1, 2. Then g1 and

g2 are isomorphic iff G1 and G2 are locally analytically isomorphic.

In the special case of one-to-one (invertible) mapping from a given PDE to a target PDE the classof possible mappings Ml is predetermined. In particular for the case on a single dependent variable,as shown in the following theorem, we have that l = 1 i.e. µ ∈M1.

Theorem 1.4.2 (Backlund(1876)). Let us consider a PDE with a single dependent variable u. Then

a mapping µ defines an invertible mapping from (x, u, . . . , u(p))-space to (y, v, . . . , v(p))-space for

any fixed p, iff µ is a one-to-one contact transformation of the form ∵y = φ(x, u, u(1))

v = ψ(x, u, u(1))

v(1) = ψ(1)(x, u, u(1))

(1.93)

Let us further notice that if φ and ψ are independent of u(1) then defines a point transformation.

A result for more that one dependent variable does exist but will not be treated in this thesis. Forthe interested reader we refer to [BK89; BCA10].2

1.4.3 Lie’s classical results

We now recall Lie’s classical results on classification of second order linear partial differentialequations. In the case of a parabolic equation it reads as follows.

Theorem 1.4.3. Let us consider the family of linear parabolic equations

P (x, t)ut +Q(x, t)ux +R(x, t)uxx + S(x, t)u = 0, P 6= 0, R 6= 0 (1.94)

The principal Lie algebra gof equation (1.94) is spanned by the generators of trivial symmetriesv1 = u∂u

vα = α(x, t)∂u(1.95)

Thus any equation of the form (1.94) can be reduced to an of equation of the form

vτ = vyy + Z(y, τ)v (1.96)

by Lie’s equivalence transformation

y = α(x, t), τ = β(t), v = γ(x, t)v, αx 6= 0, βt 6= 0 (1.97)

2It can be proved that l = 0 i.e. M0.

1.5 The Heat Equation 63

The group of dilations generated by the operator v1 reflects the homogeneity of Equation (1.94), while

the infinite group with the operator vα represents the linear superposition principle for Equation

(1.94).

If equation (1.94) admits an extension of the principal Lie algebra by one additional operator,

namely

v2 = ∂t

then the original equation reduces to the form

vτ = vyy + Z(y)v (1.98)

If furthermore g admits an extension by three additional operator, namelyv1 = ∂t

v2 = 2τ∂τ + y∂y

v3 = τ2∂τ + τy∂y −(

14y

2 + 12τ)v∂v

(1.99)

then equation (1.94) reduces to the form

vτ = vyy +A

y2v (1.100)

If g admits five additional operator

v1 = ∂t

v2 = 2τ∂τ + y∂y

v3 = τ2∂τ + τy∂y −(

14y

2 + 12τ)v∂v

v4 = ∂y

v5 = 2τ∂y − yv∂v

(1.101)

then equation (1.94) can be mapped in the heat equation

vτ = vyy (1.102)

Equations (1.98), (1.100) and (1.102) provide the canonical forms of all linear parabolic second-

order equations (1.94) that admit non-trivial symmetries.

1.5 The Heat Equation

We will now apply all the theory stated to the one-dimensional (in space) heat equation.

Theorem 1.2.15 together with the prolongation formulae (1.41) and (1.42) give a rigorous proce-dure to evaluate a symmetry group of a given partial differential equation.

Let us now finally give a concrete example of how the Lie group and the local transformation that


gives rise to it can be evaluated.

Let us thus consider the one-dimensional heat equation

ut = uxx (1.103)

We can identify the heat equation with the linear subvariety in X × U (2) determined by thevanishing of P (x, t, u(2)) = ut − uxx. Let

v = ξ(x, t, u)∂

∂x+ τ(x, t, u)

∂

∂t+ φ(x, t, u)

∂

∂u(1.104)

be a vector field on X × U . We are willing to evaluate ξ, τ, φ such that the corresponding one-parameter group exp(εv) is a symmetry group of the heat equation. By theorem 1.2.15, using thesecond prolongation

pr(2)v = v + φx(x, t, u)∂

∂ux+ φt(x, t, u)

∂

∂ut+

+ φxx(x, t, u)∂

∂uxx+ φxt(x, t, u)

∂

∂uxt+ φtt(x, t, u)

∂

∂utt

(1.105)

Applying therefore pr(2)v to (1.119) we obtain

φt = φxx (1.106)

From (1.42) we can evaluate

φt = Dt(φ− ξux − τut) + ξuxt + τutt = Dtφ− uxDtξ − utDtτ =

= φt − ξtux + (φu − τt)ut − ξuuxut − τuu2t

and

φxx = D2x(φ− ξux − τut) + ξuxxx + τuxxt

= D2xφ− uxD2

xξ − utD2xτ − 2uxxDxξ − 2uxtDxτ =

= φxx+ (2φxu − ξxx)ux − τxxut + (φuu − 2ξxu)u2x − 2τxuuxut − ξuuu3

x+

− τuuu2xut + (φu − 2ξx)uxx − 2τxuxt − 3ξuuxuxx − τuutuxx − 2τuuxuxt

(1.107)

Substituting now this two equation into (1.106) and equating ut = uxx we find the determining


equation for the symmetric group of the heat equation to be

uxtux(−2τt) (1.108a)

uxt(−2τx) (1.108b)

u2xx(τu − τu) (1.108c)

uxxu2x(−τuu) (1.108d)

uxxux(ξu − 2τxu − 2ξu) (1.108e)

uxx(τt − φu − τxx + φu − 2ξx) (1.108f)

u3xx(−ξuu) (1.108g)

u2x(−2ξxu + φuu) (1.108h)

ux(ξt + 2φxu − ξxx) (1.108i)

1(−φt + φxx) = 0 (1.108j)

(1.108k)

We have now to equate each monomial to 0. First of all let us notice that (3.34a) and (3.34b)requires that τ depends only on t. Then (3.34e) implies that ξ does not depend on u and (1.108f)implies ξ(x, t) = 1

2τtx + σ(t). Further from (1.108h) we can deduce that φ is linear in u i.e.φ(x, t, u) = β(x, t)u+ α(x, t) for some functions α and β.

For what it concerns (1.108i) ξt = −2βx so that β = − 18τttx

2 − 12σtx + ρ(t). Finally from

(1.108j) we can say that both α and β are solutions for the heat equation i.e.

αt = αxx βt = βxx

Therefore we obtainτttt = 0 σtt = 0 ρt = −1

4τtt

We can thus conclude that the most general form of the coefficients of the infinitesimal symmetry ofthe heat equation are

ξ = c1 + c4x+ 2c5t+ 4c6xt

τ = c2 + 2c4t+ 4c6t2

φ = (c3 − c5x− 2c6t− c6x2)u+ α(x, t)

Therefore we have found that the Lie algebra of infinitesimal symmetries of the heat equation is


spanned by

v1 = ∂x

v2 = ∂t

v3 = x∂x + 2t∂t

v4 = 4tx∂x + 4t2∂t − (x2 + 2t)u∂u

v5 = 2t∂x − xu∂uv6 = u∂u

together with the infinite dimensional subalgebra

vα = α(x, t)∂u

where α is an arbitrary solution of the heat equation.

The commutator table is therefore

v1 v2 v3 v4 v5 v6

v1 0 0 v1 v5 − 12 v6 0

v2 0 0 2v2 v3 − 12 v6 v1 0

v3 −v1 −2v2 0 2v4 v5 0v4 −v5 −v3 + 1

2 v6 −2v4 0 0 0v5

12 v6 −v1 −v1 0 0 0

v6 0 0 0 0 0 0

The one-parameter group Gi generated by the vi are thus given exponentiating the infinitesimalgenerator, i.e. the transformed points are exp(εvi)(x, t, u) = (x, t, u). Applying (1.48) we find

G1 : (x+ ε, t, u)

G2 : (x, t+ ε, u)

G3 : (x, t, eεu)

G4 : (eεx, e2εt, u)

G5 : (x+ 2εt, t, u exp(−εx− ε2t))

G6 :(

x1−4εt ,

t1−4εt , u

√1− 4εt exp

(−εx2

1−4εt

))Gα : (x, t, u+ εα(x, t))

Now from (1.30) and from the fact that each Gi is a symmetric group we can say that whenever


u = f(x, t) is a solution of the heat equation so are the functions

u(1) = f(x− ε, t)

u(2) = f(x, t− ε)

u(3) = eεf(x, t)

u(4) = f(e−εx, e−2εt)

u(5) = e−εx+ε2tf(x− 2εt, t)

u(6) = 1√1−4εt

exp(−εx2

1+4εt

)f(

x1+4εt ,

t1+4εt

)u(α) = f(x, t) + εα(x, t)

We would like now to recover a solution to the heat equation using the algebra just found.

Invariant solutions u = f(x, t) of the heat equation corresponding to v6 satisfy

4xt∂f

∂x+ 4t2

∂f

∂t= −(x2 + 2t)f (1.109)

We can find a solution by solving the characteristic equations

dx

4xt=

dt

4t2=

du

−(x2 + 2t)(1.110)

We thus have two invariants:X1 =

x

tv =√te

x2

4t u (1.111)

Therefore the solution of (1.109) is defined by the invariant form

√te

x2

4t u = φ(x

t) (1.112)

or solving for u

u = f(x, t) =1

te−

x2

4t φ(ζ) (1.113)

where the similarity variable isζ =

x

t(1.114)

Substituting now (1.113) into the heat equation we obtain

φ′′(ζ) = 0 (1.115)

so that the invariant solutions resulting from v6 are

u = f(x, t) =1√te−

x2

4t

(C1 + C2

x

t

)(1.116)

where C1 and C2 are arbitrary constants.

We have already found the one-parameter family of solutions u = f(x, t; ε) generated by v6 to be


G6 :

(x

1− 4εt,

t

1− 4εt, u√

1− 4εt exp

(−εx2

1− 4εt

))(1.117)

Then

u = f(x, t; ε) =1√

1− εtexp

(εx2

4(1− εt)

)f

(x

1− εt,

t

1− εt

)(1.118)

-4

-2

0

2

4

-4

-2

0

2

4

-4

-2

0

2

4

Figure 1.8: Vector field v6

We want now use the method explained to recover fundamental solutions of the heat equation viainvariance method for finite and finite domain.

We will once again consider the one-dimensional heat equation

ut = uxx (1.119)

defined on the domain t > 0

a < x < b


Recall that the determining equations for the 6-parameter Lie groups of transformations areξ = c1 + c4x+ 2c5t+ 4c6xt

τ = c2 + 2c4t+ 4c6t2

φ = (c3 − c5x− 2c6t− c6x2)u+ α(x, t)

(1.120)

From the invariance of t = 0 we have τ(0) = 0 and therefore

c2 = 0

If the domain in infinite, i.e. a = −∞ and b =∞ there is no further parameter reduction from theinvariance of the boundary surfaces. If instead a 6= −∞ then the invariance of the surface x = a

leads to ξ(a, t) = 0 for any t and hence we obtain

c1 = −c4a, 2c5 = −4c6a

Again if b 6=∞ the invariance of the surface x = b yields

c1 = −c4b, 2c5 = −4c6b

Therefore it is easy to see that if both a, b 6=∞, then c1 = c4 = c5 = c6 = 0 and thus there is no nontrivial group admitted by the BVP for the heat equation.

However, as briefly mentioned in section 1.3.3, since we are dealing with a linear PDE, it notnecessary that the Lie group of transformations leaves all the BVP invariants.

(i) Infinite domain (a, b) = (−∞,∞).

Let us consider the problemut = uxx

with boundary conditions

u(±∞, t) = 0, t > 0; u(x, 0) = δ(x)

where δ is the Dirac mass.

The infinitesimal 1.120 with c2 = 0 is admitted by the BVP provided that

φ(x, 0)u(x, 0) = ξ(x, 0)δ′(x)

when u(x, 0) = δ(x) i.e.φ(x, 0)δ(x) = ξ(x, 0)δ′(x)

The last equation is satisfied if

ξ(0, 0) = 0 and φ(0, 0) = − ∂ξ∂x

(0, 0)


implying thusc1 = 0, c3 = −c4

We have hence found that a three parameter Lie group of transformations leaves the BVPinvariant.

Infinitesimal generators of this group arev1 = x ∂

∂x + 2t ∂∂t − u∂∂u

v2 = t ∂∂x −12xu

∂∂u

v3 = xt ∂∂x + t2 ∂∂t −

[x2

4 + t2

]u ∂∂u

The matrix (1.79) is now

T =

x 2t

t o

xt t2

(1.121)

T has rank two so that the number of independent variables reduces to 0. Note further that

v3 =1

2v1 + xv2

Hence an invariant solution corresponding to v1 and v2 is also an invariance solution for v3.

Let u = f(x1) be an invariant solution corresponding to v1 and v2

v1 [u− f(x1)] = 0

leads to the invariant of the form

u = f(x1) =1√tF1(ζ1)

with similarity variable ζ1 = x√t.

The equationv2 [u− f(x1)] = 0

leads to the invariant formu = f(x1) = e−

x2

4t F2(ζ2)

with similarity variableζ2 = t

From the uniqueness of the solution of the BVP we have

1√tF1(ζ1) = e−

x2

4t F2(ζ2)


and therefore √ζ2F2(ζ2) = e

ζ214 F1(ζ1) = const = c

Hence the solution of the BVP isu =

c√te−

x2

4t

The initial condition leads toc =

1√4π

From what already said it follows

v3

[u− 1√

4πte−

x2

4t

]= 0

(ii) Semi-infinite domain (a, b) = (0,∞).

Let us now instead consider the problem

ut = uxx

with boundary conditions

u(0, t) = 0, t > 0; u(x, 0) = δ(x− x)

where δ is the Dirac mass.

A three parameter Lie group of transformations is admitted by the BVP. The invariance of thesecond surface means that

φ(x, 0)u(x, 0) = ξ(x, 0)δ′(x− x)

when u(x, 0) = δ(x− x) i.e.

φ(x, 0)δ(x− x) = ξ(x, 0)δ′(x− x)

Henceξ(x, 0) = 0 and φ(x, 0) = − ∂ξ

∂x(x, 0)

We thus must have

c4 = 0, c3 =c6x

2

4

Thus the BVP admits infinitesimal generator

v1 = xt∂

∂x+ t2

∂

∂t+

[x2

4−(x2

4+t

2

)]u∂

∂u


The corresponding invariant solution has the invariant form

u = f(x) =1√te−

x2+x2

4t F (ζ)

where F (ζ) is an arbitrary function of the similarity variable ζ = xt . Substituting the solution

so found in the heat equation (1.119) we find that F (ζ) satisfies the ODE

d2F (ζ)

dζ2=x2

4F (ζ)

and henceu = f(x) =

1√t

(Ce−

(x−x)24t +De−

(x+x)2

4t

)From the initial datum it follows that

C = −D =1√4π

leading to the well known solution of the heat equation.

Chapter 2

Fundamental solutions and transitiondensities

2.1 Preliminary results

We have previously empathized the role played by invariant solutions i.e. solution that are mappedinto themselves by a given symmetry group. The observant reader should have noticed that wementioned as well solutions that are mapped into other solutions, without requiring the solutions tobe the same. This means we can construct a given solution from a trivial one. The method we willdevelop in this chapter exploit this kind of properties.

The idea that lies behind the following method is to produce an integral transform of a solution byapplying a Lie symmetry to a trivial solution. We will use the Fourier and the Laplace transformssince they are among the most studied transformations and extensive tables does exist.

2.1.1 Group transformation of known solution

We will consider for the whole present chapter the Ito process X = Xt : t ≥ 0 solution of theSDE

dXt = f(Xt)dt+√

2σXγt dWt, X0 = x0 > 0 (2.1)

where as usual W = Wt : t ≥ 0 is a standard Brownian motion.

The process (2.1) can be link to the PDE of the form

ut = σxγuxx + f(x)ux − g(x)u (2.2)

where σ > 0 and γ a constant.

The present method to construct exact solution is based on the fact that a symmetry group

73

74 2. Fundamental solutions and transition densities

transforms any solutions of the equation into solution of the same equation. Namely, letx = Xε(x, u)

u = Uε(x, u)(2.3)

be a Lie group of transformations of equation

F (x, u(m)), (2.4)

and let a functionu = f(x) (2.5)

solve equation (2.4). Since (2.3) is a symmetry transformation, the solution (2.5) can be also writtenusing the new variables

u = f(x)

Replacing here x, u from equations (2.3), we get

Uε(x, u) = f(Xε(u, x))

Having solved this equation with respect to u, we arrive at the following one-parameter family of newsolutions of equation (2.4))

u = Uε(x)

Consequently, any known solution is a source of a multi-parameter class of new solutions providedthat the differential equation considered admits a multi-parameter symmetry group.

As an example see section ?? where the symmetric group for the heat equation is given.

2.1.2 Fundamental solutions

Investigation of initial value problems for parabolic linear partial differential equations (hyperbolicas well but we will not consider the case in the thesis) can be reduced to the construction of a particularsolution with specific singularities known in the literature as fundamental solutions. What we aredoing here is just a brief introduction to fundamental solutions. For some general notions onfundamental solutions for parabolic equations see appendix B.1.3 or for a more interested reader werefer to Friedman [Fri64] and Evans [Eva10].

We now state the definition of fundamental solution for a given PDE. First of all a Cauchy problemis a PDE together with a boundary condition as followsut = a(x, t)uxx + b(x, t)ux + c(x, t)u, x ∈ Ω ⊆ RN

u(x, 0) = φ(x)(2.6)

Definition 2.1.1. A fundamental solution for the Cauchy problem (2.6) is a function p(x, y, t) such

that:

2.1 Preliminary results 75

• ∀ y fixed, p(x, y, t) is a solution of (2.6) on Ω× (0, T ] for some T > 0;

• u(x, t) =∫

Ωφ(y)p(x, y, t)dy is a solution of (2.6).

The fundamental solution is thus the solution of the following Cauchy problemut = σxγ + f(x)ux − g(x)u, t < t0

u(x, t)|t→t0 δ(x− y)(2.7)

where δ(x − y) denotes the Dirac mass at the point y. In general we will consider t0 = 0. Theconnection between definition 2.1.1 and the Cauchy problem (2.7) can be seen via convolution.

The idea is to find a fundamental solution of equation (2.2). If the fundamental solution of thisPDE satisfies certain conditions, then it gives the probability density function for the process X .

What we will be doing is starting from (2.2), using the method previously exposed, we will finda basis for the Lie algebra. Therefore composing properly this basis we will find an infinitesimalgenerator such that the related symmetric group Uε(t, x) possesses certain properties.

Now let us suppose that (2.2) has a fundamental solution p(x, y, t), therefore the function

u(x, t) =

∫Ω

φ(y)p(x, y, t)dy (2.8)

has to solve the initial value problem (2.6).The main idea of the method is to connect (2.8) with the symmetric group Uε(t, x).Taking u0 = 1 stationary solution we will often be able to recover the transition density function

of the process to be studied. Thus from definition 2.1.1, setting thus t = 0 and recalling that Uε(t, x)

is as matter of a fact a solution of (2.2) we have the following∫Ω

Uε(0, x)p(x, y, t)dy = Uε(t, x) (2.9)

The key point is that we have to choose ε such that the the family of solutions Uε(x) reducesto a Laplace or a Fourier transform when setting t = 0. This will allow us to invert (2.9) and thento recover the transition density. The inversion does not imply p(x, y, t) to be a transition density.However we will see that some theorems exist such that this is true.

Let us now give an example in order to make the method more clear.

Example 2.1.1. We have previously found the basis for the Lie symmetry algebra for theone-dimensional heat equation.

We have in fact seen in section ?? that if u(x, t) solves the heat equation so does u(5) =

exp(−εx+ ε2t)u(x− 2εt, t). Taking now as said u0 = 1 we will have∫Re−εyp(t, x− y)dy = exp(−εx+ ε2t)

This is as we wanted a Laplace transform and we can thus recover p(t, x− y) inverting it.


2.1.3 Integrating symmetries

Suppose we are dealing with a PDE of the form

F (x, u(m)) =∑|α|≤m

aα(x)Dα, x ∈ Ω ⊆ RN (2.10)

with α = (α1, . . . , αN ), αi ∈ N, |α| = α1 + · · ·+ αN and Dα = ∂|α|

∂xα11 ...∂x

αpp

.

Let now G be a one parameter group of transformations, generated by the vector field

v =

N∑i=1

ξ(x)∂xi + φ(x, u)∂u

As usual we will denote the action of G on a solution u by exp(εv)(u(x)) = (exp εv)u(x) = Uε(x).Thus by the Lie symmetry property it exists an interval I ⊆ R containing the identity e = 0 such thatUε(x) is a continuous one parameter family of solutions of (2.10). From continuity and linearity wehave the following

Lemma 2.1.2. Suppose Uε(x) is a continuous one parameter family of solution of the PDE (2.10).Suppose furthermore the ϕ : I → R is a function with sufficiently rapid decay. Then

u(x) =

∫I

ϕ(ε)Uε(x)dε

is a solution of (2.10). Furthermore if the PDE is time dependent we can define Uε(x, t) the family of

symmetry solutions to have

u(x, t) =

∫I

ϕ(ε)Uε(x, t)dε and u(x, 0) =

∫I

ϕ(ε)Uε(x, 0)dε.

FurtherdnUε(x)

dεn

is also a solution for all n = 0, 1, 2, . . . .

Proof. It is enough to differentiate under the integral. If the interval I is unbounded then the functionϕ must be of compact support in order to have the convergence of the integral. Eventually the lemmais proved noticing that the PDE does not depend on ε so that the order of differentiability may bereversed.

2.1.4 Fundamental solutions as transition densities

In this section we will provide a link between SDE and PDE. In particular we will show how touse the previous result studying a PDE. The importance of fundamental solution is that they are thelink from analysis to probability theory (and viceversa).

2.1 Preliminary results 77

Let us consider an Ito diffusion X = Xt : t ≥ 0 which satisfies the SDEdXt = µ(Xt, t)dt+ σ(Xt, t)dWt

X0 = x(2.11)

where as usual Wt denotes a standard Wiener process. Let us assume that µ and σ satisfy certainconditions such that a unique solution does exist. Then

u(x, t) = Ex[φ(Xt)] := E[φ(Xt)|X0 = x] (2.12)

is a solution of the Cauchy problemut = 12σ

2(x, t)uxx + µ(x, t)ux,

u(x, 0) = φ(x)(2.13)

The PDE (2.13) is known as Kolmogorov forward equation. We will not enter in this thesisthe discussion about well-posedeness and ill-posedeness of parabolic equation. We will just recallthat when dealing with PDE’s in financial mathematics describing the evolution of the value of agiven option, the datum is prescribed at maturity. This implies that the forward parabolic equation isill-posed and thus a change of variable τ = T − t with T maturity date has to be made.

Back to the beginning, if p(x, y, t) is an appropriate fundamental solution of (2.13), we cancompute the expectation according to definition 2.1.1 and equation (2.12) u(x, t) = Ex[φ(Xt)] =∫

Ωφ(y)p(x, y, t)dy. In a probability context the fundamental solution is known as transition density

function. Of course in order p(x, y, t) to be a density function we have to require∫

Ωp(x, y, t)dy = 1.

Fundamental solutions are not unique. The PDE (2.13) may have more than one fundamentalsolution, and of course only one of them can be the desired transition density function. The strength ofthe Lie algorithm is that we are able to recover the fundamental solution that gives us the probabilitydensity function (pdf).

Example 2.1.2. Let us consider the diffusion X = Xt : t ≥ 0 which satisfies the SDEdXt = 2aXt2+aXt

+√

2XtdWt

X0 = x > 0, a > 0(2.14)

Since the drift is Lipschitz continuous for a > 0 adn√x is Holder continuous by the Yamada-

Watanabe theorem the SDE (2.14) has a unique strong solution. We thus look for a fundamentalsolution of

ut = xuxx +2ax

2 + axux (2.15)

It can be shown that equation (2.15) has a fundamental solution of the form

q(x, y, t) =2 + ay

t(2 + ax)

√x

yexp−x+ y

tI1(

2√xy

t

)(2.16)


The problem is that the fundamental solution (2.16) cannot be the pdf. In fact let us notice that∫ ∞0

q(x, y, t)dy = 1−exp− x

2t2 + ax

6= 1

In fact it has been shown in [CP04] that the transition probability function for (2.14) is

p(x, y, t) =exp−x+y

t t(2 + ax)

[√x

y(2 + ay)I1

(2√xy

t

)+ tδ(y)

]

In general in order to obtain a pdf it is often necessary to include additional terms which involvegeneralized function such as the Dirac mass. The method we develop in the present chapter gives usthese additional terms that cannot otherwise be found.

A more general result that emphasize once more the connection between SDE and parabolic PDEis the Feynman-Kac theorem.

Theorem 2.1.3 (Feynman-Kac). Let u ∈ C2(ST ) ∩ C(ST ) be a solution of the Cauchy problemut + 12σ

2uxx + µux − ru+ f = 0, in ST := [0, T )× Rm

u(x, 0) = φ(x)

Let X = Xt : t ≥ 0 be an Ito diffusion process which satisfies the SDEdXt = µ(Xt, t)dt+ σ(Xt, t)dWt

X0 = x(2.17)

Let us now assume that:

(i) µ and σ have at most linear growth in x;

(ii) ∀ (t, x) ∈ ST , ∃ a solution X of the SDE (2.17);

Let us assume furthermore that at least one of following holds:

1. ∃ two constants M,p > 0 :

|u(t, x)|+ |f(t, x)| ≤M(1 + |x|p) (t, x) ∈ ST

2. the matrix σ is bounded and ∃ two constants M,α > 0, with α small enough :

|u(t, x)|+ |f(t, x)| ≤M(expα|x|2

) (t, x) ∈ ST

2.2 The square root process 79

Then ∀ (t, x) ∈ ST we have the following formula

u(x, t) = Et,x

[exp

(−∫ T

t

r(u,Xu)du

)φ(XT ) +

∫ T

t

exp

(∫ u

t

r(z,Xz)dz

)f(u,Xu)du

](2.18)

2.2 The square root process

The present section is devoted to some general results that give us the explicit formulation ofsolution for a wide class of PDE’s. We will proceed as explained in the previous section. Eventuallywe will use these results in chapter 3.

2.2.1 General results

As said we are trying to invert via Laplace transform a fundamental solution. Such a fundamentalsolution in in general a distribution (or generalized function). Let us first of all introduce thegeneralized Laplace transform for distributions.

Definition 2.2.1 (Generalized Laplace transform). Let f : [0,∞)→ R be Lebesgue integrable and

be of suitable slow growth. The generalized Laplace transform of f is the function

Fγ(λ) =

∫ ∞0

f(y)e−λy2−γ

dy

where λ > 0 and γ 6= 2.

This transform is usually inverted setting z = y2−γ in order to deal with a Laplace transforms.

Proposition 2.2.2. The PDE

ut = σxγuxx + f(x)ux − g(x)u, γ 6= 2 (2.19)

has a Lie algebra of symmetries which is at least four dimensional iff, for a given function g, the

function h(x) = x1−γf(x), is a solution of one of the following families of Riccati equations:

σxh′ − σh+1

2h2 + 2σx2−γg(x) = 2σAx2−γ +B

σxh′ − σh+1

2h2 + 2σx2−γg(x) =

Ax4−2γ

2(2− γ)2+Bx2−γ

2− γ+ C

σxh′ − σh+1

2h2 + 2σx2−γg(x) =

Ax4−2γ

2(2− γ)2+Bx3− 3

2γ

3− 32γ

+Cx2−γ

2− γ−K

with K = γ8 (γ − 4)σ2, and A,B and C arbitrary constants.

Symmetry solutions can be found for h that satisfies any of the three previous equations. Anyhowwe will state only results concerning the first and second Riccati equation.


Thus we can find fundamental solution inverting a generalized Laplace transform. we now statean important result that we will need when trying to recover fundamental solutions.

Proposition 2.2.3. When n is a non negative integer we have

L−1[λne

kλ

]=

n∑l=0

kl

l!δ(n−l)(y) +

(k

y

)n+12

In+1(2√ky)

where L is the Laplace transform, δ(y) is the Dirac delta and In is a modified Bessel function of the

first Kind.

Hopefully we are now able to recover fundamental solutions for certain PDE’s. We are still notable to say anything about this fundamental solution to be a probability transition density. We willnow look for certain condition under which we can say that the fundamental solution found is in facta transition density.

In general we are able to recover a pdf in the case of g = 0. For the more general situation ofg 6= 0 some problem may arise.

Corollary 2.2.4. Suppose that the conditions of 2.2.10 hold and that g = 0. Let us take the

stationary solution u0 = 1. The resulting fundamental solution p(x, y, t) has the property that∫∞0p(x, y, t)dy = 1.

This result is still not enough to say that the fundamental solution is a transition density. In orderto do this we need the following results.

Proposition 2.2.5. Let X = Xt : t ≥ 0 be an Ito process solution of

dXt = f(Xt)dt+√

2σXtdWt

with |f(x)| ≤ Keax measurable. It therefore exists T : u(x, t, λ) = Ex[e−λXt ] is the unique solution

of

ut + λ2σuλ + λEx[f(Xt)e−λXt ] = 0

subject to u(x; 0, λ) = e−λx for 0 ≤ t < T , λ > a.

Corollary 2.2.6. Let X = Xt : t ≥ 0 be the unique strong solution of the SDE

dXt = f(Xt)dt+√

2σXγt dWt, X0 = x0 > 0

and suppose that h(x) = x1−γf(x) is a solution of either of the Riccati equations. Then

Ex[e−λXt

]= Uλ(x, t)

where the value of Uλ(x, t) is given by taking u0 = 1

We have now all the necessary tools to properly a wide class of partial differential equations.


2.2.2 First Riccati equation

When dealing with PDE’s whose drift satisfies the first of the Riccati equation we have in generalgood analytic results. The main result for this class of processes is the following.

Theorem 2.2.7. Suppose γ 6= 0 and h(x) = x1−γf(x) is a solution of the first Riccati equation

σxh′ − σh+1

2h2 + 2σx2−γg(x) = 2σAx2−γ +B

Then the PDE (2.19) has a symmetry solution of the form

Uε(x, t) =1

(1 + 4εt)1−γ2−γ

exp

[−4ε(x2−γ +Aσ(2− γ)2t2)

σ(2− γ)2(1 + 4εt)

]× (2.20)

× exp

[1

2σ

(F

(x

(1 + 4εt)2

2−γ

)− F (x)

)]u

(x

(1 + 4εt)2

2−γ,

t

1 + 4εt

)

with F ′(x) = f(x)xγ and u is a solution of the PDE. Thus for sufficiently small ε Uε is a solution of

(2.19) whenever u is. If u(x, t) = u0(x) with u0 an analytic, stationary solution then there is a

fundamental solution p(x, y, t) of (2.19) such that∫ ∞0

e−λy2−γ

u0p(x, y, t)dy = Uλ(x, t)

where Uλ(x, t) = U 14σ(2−γ)2λ.

Further if u0 = 1 then∫∞op(x, y, t)dy = 1.

Proof. Using the Lie algorithm we can see that (2.19) has an infinitesimal symmetry of the form

v =8xt

2− γ∂x + 4t2∂t −

(4x2−γ

σ(2− γ)2+

4x1−γtf(x)

σ(2− γ)2+ βt+ 4At2

)u∂u

where β = 4(1−γ)2−γ . Exponentiating this symmetry we will get Uε in (2.20).

A solution Uλ(x, t) should be obtained as

Uλ(x, t) =

∫ ∞0

Uλ(y, 0)p(x, y, t)dy =

∫ ∞0

eλy2−γ

u0(y)p(x, y, t)dy

We thus need to show that Uλ is the generalized Laplace transform of some distribution u0p. Since

∫ ∞0

eλy2−γ

u0(y)p(x, y, t)dy =

∫ ∞0

eλzu0(z1

2−γ )p(x, z1

2−γ , t)z

12−γ−1dz

2− γ

Thus Uλ(x, t) can be written as a product of λν for same value ν and analytic function G(1/λ). Anyfunction which is analytic in 1/λ is a Laplace transform. Further λν is a Laplace transform. HenceUλ(x, t) is a Laplace transform.

If we integrate a test function ϕ(λ) with sufficient rapid decay against Uλ(x, t) then u(x, 0) =


∫∞0Uλ(x, 0)ϕ(λ)dλ is a solution of (2.19). Further

u(x, 0) =

∫ ∞0

Uλ(x, 0)ϕ(λ)dλ =

∫ ∞0

u0(x)e−λx2−γ

ϕ(λ)dλ = u0(x)ς(x)

where ς is the generalized Laplace transform of ϕ. Now by Fubini’s theorem∫ ∞0

u0(y)ςy(y)p(x, y, t)dy =

∫ ∞0

∫ ∞0

u0(y)ϕ(λ)p(x, y, t)e−λx2−γ

dλdy =

∫ ∞0

∫ ∞0


dydλ =∫ ∞0

(y)ϕ(λ)Uλ(x, t)dx = u(x, t)

But u(x, 0) = u0(x)ςx(x). Therefore integrating initial data u0ςγ against p, the resulting functionsolves the Cauchy problem for (2.19). This proves that p is a fundamental solution.

The previous theorem gives us strong results when a PDE of the form (2.19) with g = 0 isconsidered. But it is often the case that we are dealing with equations with g different from 0. Wewill now state some results for such a more general case. We will treat only the case of some specificfunction g that may arise in finance. Results on different functions can be found in [Cra09; CL09;CL07]. Anyway specific case are to be treated independently.

Setting g(x) = µ changes some of the results we have previously introduced. First of all wecannot any more take u0 = 1 as the stationary solution. How do we find then the right stationarysolution? Recall that the stationary solution is not unique. Furthermore different stationary solutionswill in general lead to different fundamental solutions. We have first of all find a way to choose theright stationary solution u0 in order to be able to compute the desired expectatin through theorem2.1.3. We will show how the choice of u0 is made using an example.

Example 2.2.1. Let us consider the process X = Xt : t ≥ 0 wheredXt = aXtdt+ dWt

X0 = x

This is better know as Bessel process. We use this process since its transition density is known. Usingthe usual notation we have σ = 1

2 , γ = 0 and f(x) = ax . Setting thus h(x) = a we have A = 0 in

the Riccati equation. Our aim is to evaluate

Ex[eµ4

∫ t0dsX2s φ(Xt)

]Therefore we have to set g(x) = µ

4x2 in order to study the PDE

ut =1

2uxx +

a

xux −

µ

4x2u


Finding a fundamental solution of this PDE we will be able to evaluate the desired expectation.

We require a fundamental solution that reduce to the known transition density of the process atµ = 0. Computing the stationary solution we find u0 = xd where d = 1

2 − a+√

µ2 + (a− 1

2 )2 tobe the right one. The discrimination point is that as µ→ 0 u0 → 1. Craddock proved in Craddockand Lennox [CL09] that if µ = 0 the transition density in fact reduces to the transition density of theBessel process.

We have then found a way to choose the right stationary solution. Anyway we have not saidanything about how to recover these stationary solutions.

Given the function h(x), in order to obtain the stationary solution u0 we need to solve

σxγuxx + f(x)ux − g(x)u = 0

We divide by xγ−1 to be able to rewrite the PDE as

σxuxx + h(x)ux − x1−γg(x)u = 0

If u(x) = u(x)e−12σ

∫ h(x)x dx then 2σ2x2u′′(x)− (2σAx2−γ +B)u(x) = 0. Finally we get through

the substitution z = x2−γ , u(x) = w(x2−γ) the following ODE

2σ2(2− γ)2z2w′′(z) + 2σ2(2− γ)(1− γ)zw′(z)− (2σAz +B)w(z) = 0 (2.21)

This equation has always Bessel functions solutions.

Lemma 2.2.8. The PDE

ut = σxuxx + f(x)ux −µ

xu

with f solution of the Riccati equation

σxf ′ − σf +1

2f2 = Ax+B

has a stationary solution uµ0 (x) which has the property that u00(x) = 1

Theorem 2.2.9. Suppose that f is a solution of

σxf ′ − σf +1

2f2 = Ax+B

Then the PDE

ut = σxuxx + f(x)ux −µ

xu, 2B + σ2 + 4µσ > 0

has a fundamental solution

p(x, y, t) =

√x

σte−

F (x)2σ −

(x+y)σt −

At2σ

1

uµ0 (y)× (2.22)


×(c1Iν

(2√xy

σt

)Iν

(√Ay

σ

)+ c2I−ν

(2√xy

σt

)I−ν

(√Ay

σ

))where F ′ = f(x)

x and ν = 1σ

√2B + σ2 + 4σµ satisfies |ν| < 1. From which we may calculate

Ex[e−λXt−µ

∫ t0dsXs

]=

∫ ∞0

e−λyp(x, y, t)dy

Proof. Let f(x) = 2σ y′

y where y(x) =√x(c1Iα

(√2Ayσ

)+ c2I−α

(√2Ayσ

)).

Using the substitution u0 = e−F (x)2σ v(x) we have that v has to satisfy

2σx2v′′(x)− (Ax+B + 2σµ)v(x) = 0

which has solution√xI±ν

(√2Axσ

). Setting now

uµ0 (x) =√xe−

F (x)2σ

(c1Iν

(√2Ay

σ

)+ c2I−ν

(√2Ay

σ

))We then have uµ0 (y) = 1. Now ∫ ∞

0

e−λyuµ0 (y)p(x, y, t)dy =

=

√xe−

F (x)2σ −

λ2(x+At2

21+λσt

1 + λσ×

×

(c1Iν

( √2Ax

σ(1 + λσt)

)+ c2I−ν

( √2Ax

σ(1 + λσt)

))Using now the fact that

L−1

[1

λexp

(m2 + n2

λ

)Id

(2mn

λ

)]= Id(2m

√y)Id(2n

√y), d > −1

Inverting the Laplace transform we recover the fundamental solution. The expectation is found viathe Feynman-Kac theorem 2.1.3

2.2.3 Second Riccati equation

As said the case of the first Riccati equation is in general analytically good. More complicated isthe case of the second Riccati equation. The structure is as previously. We will first state a results forthe case g = 0. Then we will consider some particular function g.

Theorem 2.2.10. Let us consider the PDE

ut = σxγuxx + f(x)ux − g(x)u (2.23)


and suppose that g and h(x) := x1−γf(x) satisfy the second Riccati equation

σxf ′ − σf +1

2f2 =

A

2(2− γ)2x4−2γ +

B

2− γx2−γ + C (2.24)

Let u0be a stationary, analytic solution of (??). Then (??) has a symmetric solution of the form

Uε(x, t) = (1 + ε2(cosh(√At− 1) + 2ε sinh(

√A))−c× (2.25)

×

∣∣∣∣∣cosh(√At2 ) + (1 + 2ε) sinh(

√At2 )

cosh(√At2 )− (1− 2ε) sinh(

√At2 )

∣∣∣∣∣B

2σ√A(2−γ)

exp

(1

2σF (x)− Bt

2σ(2− γ)

)×

× exp

(−√Aεx2−γ(cosh(

√At) + ε sinh(

√At))

σ(2− γ)2(1 + ε2(cosh(√At)− 1) + 2ε sinh(

√At))

)×

× exp

(1

2σF

(x

(1 + ε2(cosh(√At)− 1) + 2ε sinh(

√At))

12−γ

))×

×u0

(x

(1 + ε2(cosh(√At)− 1) + 2ε sinh(

√At))

12−γ

)×

with F ′(x) = f(x)xγ and c = 1−γ

2−γ .

Furthermore it exists a fundamental solution p(x, y, t) of (??) :∫ ∞0

e−λy2−γ

u0p(x, y, t)dy = Uλ(x, t)

where Uλ(x, t) = Uσ(2−γ)2λ√A

(x, t).

Proof. It can be shown that if f satisfies the condition than a basis for the symmetric group is spannedby

v1 =√Ax

2−γ e√At∂x + e

√At∂t −

(Ax2−γ

2σ(2−γ)2 +√Ax1−γ

2σ(2−γ)f(x)e√Atu∂u − αue

√At∂u

)v2 =

√Ax

2−γ e−√At∂x + e−

√At∂t −

(Ax2−γ

2σ(2−γ)2 −√Ax1−γ

2σ(2−γ)f(x)+β e−√Atu∂u

)v3 = ∂t

v4 = u∂u

where we have denoted by α = (1−γ)2(2−γ)

√A+ B

2σ(2−γ) and β = − (1−γ)2(2−γ)

√A+ B

2σ(2−γ) .

The key is to compose the basis such that when setting t = 0 the symmetric group reduces to ageneralized Laplace transform. This happens to be

v =2√Ax

2− γsinh(

√At)∂x + 2(cosh(

√At)− 1)∂t −

1

σg(x, t)u∂u


with

g(x, t) =

(Ax2−γ

(2− γ)2+

B

2− γ

)cosh(

√At) +

√A sinh(

√At)

(x1−γ

2− γf(x) + σc

)Exponentiating as explained before gives the symmetric group Uε(x, t) in (2.25).

The change of parameter ε 7→ σ(2−γ)2λ√A

implies Uλ(x, 0) = e−λx2−γ

is the wanted generalizedLaplace transform.

Now if p(x, y, t) is a fundamental solution of (2.23) then∫ ∞0

= e−λx2−γ

u0(y)p(x, y, t)dy = Uλ(x, t) (2.26)

This anyway has to be proven.

Since the drift is an analytic function then Uλ can be written as a product of functions analytic in1λ and any function analytic in 1

λ is a Laplace transform.

Let us now suppose (2.26) holds for some distribution p. We have to prove that p is a fundamentalsolution for the PDE. The fact that Uλ(x, t) together with (2.26) implies p is a solution for any fixedy. Integrating now a test function ϕ(λ) with sufficiently rapid decay against Uλ. Then the functionu(x, t) =

∫∞0Uλ(x, t)ϕ(λ)dλ is a solution of (2.2) from lemma 2.1.2.

Furthermore we have

u(x, 0) =

∫ ∞0

Uλ(x, 0)ϕ(λ)dλ =

∫ ∞0

u0(x)e−λx2−γ

ϕ(λ)dλ = u0(x)ςγ(x)

where ςγ is the generalized Laplace transform of ϕ. Further by Fubini’s theorem∫ ∞0

u0(y)ςγ(y)p(x, y, t)dy =

∫ ∞0

∫ ∞0


dλdyϕ(λ)dλ =

∫ ∞0

∫ ∞0


dydλϕ(λ)dλ =∫ ∞0

(y)ϕ(λ)Uλ(x, t)dx = u(x, t)

But u(x, 0) = u0(x)ςγ(x). Therefore integrating initial data u0ςγ against p, the resulting functionsolves the Cauchy problem for (2.23). This proves that p is the fundamental solution.

Again for the particular case of g = 0 a strong result is stated in theorem (2.2.10). What can besaid in stead for the general case of a non null function g?

The problem when dealing with a solution of the second Riccati equation is that the criterion forsearching fundamental solutions we have introduced before in lemma 2.2.8 does not apply here. Infact it is not necessary that a stationary solution does not reduce to 1 when setting µ = 0. We havethus to find a different approach from the inversion via Laplace transform.

An alternative approach to the construction of the necessary fundamental solution is to use groupinvariant solutions. We mean by group invariant solution is a solution which is left invariant under the


action of a group of transformation. We now need definition 1.3.13 and theorem 1.3.14 in order to beable to state the next result.

Anyway we can say that when dealing with an equation of the second family of the Riccatiequations we always have a solution of the form

w(z) = xβe−√Ax

2σ

(c11F1(α, 2β,

√Ax

σ) + c2U(α, 2β,

√Ax

σ)

)

where α =B√A

+σ+σ√

2Cσ2

+1

2σ , β = 12

(√2Cσ2 + 1 + 1

), 1F1(a, b, z) = M(a, b, z) =

∑∞n=0

a(n)zn

b(n)n!

with a(n) = a(a + 1) . . . (a + n − 1) is the Kummer’s confluent hypergeometric function andU(a, b, z) = Γ(b−1)

Γ(a) z1−b1F1(a− b+ 1, 2− b, z) + Γ(1−b)

Γ(a−b+1) 1F1(a, b, z) is the Tricomi’s confluenthypergeometric function.

Theorem 2.2.11. Let us suppose f is a solution of the second Riccati equation

σf +1

2f2 + 2σµx2 =

1

2Ax2 +Bx+ C, with A > 0 (2.27)

Then ∃ a fundamental solution of

ut = σxuxx + f(x)ux − µxu, x ≥ 0 (2.28)

of the form

p(x, y, t) =

√Axye−(F (x)−F (y)/(2σ))

2σ sinh(√At/2)

exp

(−Bt

2σ−

√A(x+ y)

2σ tanh(√At/2)

)(C1(y)Iν

( √Axy

σ sinh(√At/2)

)+ C2(y)I−ν

( √Axy

σ sinh(√At/2)

)) (2.29)

where ν =√σ2+2Cσ and F ′(x) = f(x)

x .

Proof. Whenever the drift f satisfies the given Riccati equation, then a basis for the symmetric groupis given by

v1 = xe√At∂x + e

√At√A∂t − 1

2σ

(√Ax+ f(x) + B√

A

)e√Atu∂u

v2 = −xe−√At∂x + e−

√At

√A∂t − 1

2σ

(√Ax− f(x) + B√

A

)e−√Atu∂u

v3 = ∂t

v4 = u∂u

vβ = β(x, t)∂u

β is an arbitrary solution of the PDE. Obviously adding this will not give us an invariant solution. Wetherefore look for symmetries of the form v =

∑4i=1 ckvk which preserves the boundary conditions.

We require x = 0 to be preserved by the action of v. That is v(x) = 0 when x = 0. At thesame time we require the boundary t = 0 to be preserved. As before this is v(t) = 0 when t = 0.


The last boundary condition u(x, 0) = δ(x− y) must be preserved that is again v(x− y) = 0 whenu(x, 0) = δ(x − y). In order this conditions to be satisfied we need c1 = c2, c3 = − 2√

Ac1 and

c4 = 2 (Ay+B)

σ√Ac1.

Therefore the action we are looking for will be of the form

v = c3

(− 2√

Av3 + v1 + v2 + 2

(Ay +B)

σ√A

v4

)Invariants of the action, as explained in the first part, are found solving v(η) = 0. Thay can be foundthrough the method of characteristic. We can thus consider η and v where

η =x

4 sinh2(√

At/2) (2.30)

u = exp

(−Bt+ F (x)− F (y)

2σ−

√A(x+ y)

2σ tanh(√At)/2

)v

x

4 sinh2(√

At/2) (2.31)

Now v(η) must satisfy 2σ2η2v′′(η)− (C + 2Ayη)v(η) = 0 when u is a solution of the PDE. Hence

v(η) =√yη

(C1(y)Iν

(2√Ayη

σ

)+ C2(y)I−ν

(2√Ayη

σ

))since the solution is a group invariant solution for the action generated by v the result follows fromsubstituting v and η into (2.31).

Chapter 3

Financial markets

3.1 The Cox-Ingersoll-Ross (CIR) model

The CIR model is one of the most used process in finance. It is usually used to describe thebehaviour of interest rate. It has several properties that make such a process the ideal SDE whendealing with financial markets. For an extensive treatment of such a model we refer to Bri;Cox.

3.1.1 CIR transition density function

We skip any financial topic and we will focus on the CIR process and on its symmetric group.

We will study the stochastic process X = Xτ : τ ≥ 0 solution of the CIR model, see CIR fordetails, described by the following SDEdXτ = k(θ −Xτ )dτ + σ

√XτdWτ

X0 = x(3.1)

We want to recover the fundamental solution associated to the SDE (3.1) exploiting the methoddescribed previously.

In particular the PDE associated to equation (3.1) is of the form described by equation (2.23), i.e.it reads as follows

uτ =σ2

2xuxx + k(θ − x)ux . (3.2)

According to notation introduced in Th. (2.2.10), we have

γ = 1 ; g(x) = 0 ; f(x) = k(θ − x) ; h(x) = f(x) = k(θ − x) ,

and in order h(x) to satisfy an equation of type (2.24) we obtain

−σ2

2xk +

σ2

2xk − σkθ +

(kθ)2

2+k2x2

2− k2θx =

A

2x2 +Bx+ C ,

89

90 3. Financial markets

if and only if parameters A,B,C satisfy

A = k2 ; B = −k2θ , C =kθ

2(kθ − σ2) .

We can finally apply (2.2.10) with F (x) = kθ ln(x) − kx and u0(x) = 1 to obtain the desiredsolution, namely

Uσ2(2−γ)2λ2√A

(x, τ) =2k

kθσ2 e

2k2θτσ2

(σ2

2 λ(ekτ − 1) + kekτ )2kθσ2

exp

(−λkx

(σ2


),

in fact we have that

Uε(x, τ) =

∣∣∣∣∣ cosh( kτ2 ) + (1 + 2ε) sinh( kτ2)

cosh( kτ2)− (1− 2ε) sinh( kτ

2)

∣∣∣∣∣−kθσ2

exp

(−

1

σ2(kθ ln(x)− kx) +

2k2θτ

σ2

)×

× exp

(−kεx(cosh(kτ) + ε sinh(kτ))

σ2

2(1 + 2ε2(cosh(kτ)− 1) + 2ε sinh(kτ))

)exp

(kθ

σ2ln

(x

1 + 2ε2(cosh(kτ)− 1) + 2ε sinh(kτ)

))×

× exp

(−kx

σ2

(1


))=

=

∣∣∣∣∣ cosh( kτ2 ) + (1 + 2ε) sinh( kτ2)

cosh( kτ2)− (1− 2ε) sinh( kτ

2)

∣∣∣∣∣−kθσ2

ek2θτσ2 exp

[−kθ

σ2log

(x(1 + ε2(cosh(kτ)− 1) + 2ε sinh(kτ))

x

)]︸︷︷︸

≡Part1

×

× exp

(kx

σ2

1− 2ε(cosh(kτ)− ε sinh(kτ)


)︸︷︷︸

≡Part2

.

(3.3)

Let us focus our attention separately on Part1 and Part2 of the equation (3.3). For what concernsPart1 we have

Part1 =

∣∣∣∣∣cosh(kτ2 ) + (1 + 2ε) sinh(kτ2 )

cosh(kτ2 )− (1− 2ε) sinh(kτ2 )

∣∣∣∣∣−kθσ2

ek2θτσ2 [1 + 2ε2(cosh(kτ)− 1) + 2ε sinh(kτ)]

−kθσ2 =

=

∣∣∣∣∣∣ekτ2 + e

−kτ2 + (1 + 2ε)

(ekτ2 − e−kτ2

)ekτ2 + e

−kτ2 − (1− 2ε)

(ekτ2 − e−kτ2

)∣∣∣∣∣∣−kθσ2

ek2θτσ2 [1 + ε2(ekτ + e−kτ − 2) + ε(ekτ − e−kτ )]

−kθσ2 =

=

∣∣∣∣∣ekτ2 2ε+ e

−kτ2 (2− 2ε)

ekτ2 (2 + 2ε)− e−kτ2 2ε

∣∣∣∣∣kθσ2(

1

ekτ (ε2 + ε) + e−kτ (ε2 − ε)− 2ε2 + 1

) kθσ2

ek2θτσ2 =

=

(ekτ

(ε− ekτ (1 + ε))2

) kθσ2

ek2θτσ2 ,

hence, taking ε = σ2λ2k we get Part1 =

(2kekτ

σ2λ(ekτ−1)+2kekτ

) 2kθσ2

.

Concerning the second part of equation (3.3), we have

3.1 The Cox-Ingersoll-Ross (CIR) model 91

Part2 = exp

(kx

σ2

(1 + 2ε2(cosh(kτ)− 1) + 2ε sinh(kτ)− 1− 2ε(cosh(kτ)− 2ε2 sinh(kτ))


))=

= exp

(kx

σ2

(2ε2 (cosh(kτ)− 1) + 2ε sinh(kτ)− 2ε cosh(kτ)− 2ε2 sinh(kτ)


))=

= exp

kxσ2

2ε2(ekτ+e−kτ

2 − 1)

+ 2ε(ekτ−e−kτ

2

)− 2ε

(ekτ+e−kτ

2

)− 2ε2

(ekτ−e−kτ

2

)1 + 2ε2(cosh(kτ)− 1) + 2ε sinh(kτ)

=

= exp

kxσ2

ekτ (ε2 − ε2 + ε− ε) + e−kτ (2ε2 − 2ε)− 2ε2

1 + 2ε2(ekτ+e−kτ

2 − 1)

+ 2ε(ekτ−e−kτ

2

) =

== exp

(kx

σ2

(e−kτ (2ε2 − 2ε)− 2ε2

ekτ (ε2 + ε) + e−kτ (ε2 − ε)− 2ε2 + 1

))= exp

(−2kxε

−σ2ε+ ekτσ2(1 + ε)

),

thus, taking ε = σ2λ2k , we obtain Part2 = exp

(−2kxλ

σ2λ(ekτ−1)+2kekτ

).

Combining Part1 and Part2, we finally have

Uσ2(2−γ)2λ2√A

(x, τ) =

∫ ∞0

e−λyp(x, y, τ)dy =

=k

2kθσ2 e

2k2θτσ2

(σ2


exp

(−2λkx

σ2λ(ekτ − 1) + 2kekτ

),

which is nothing but the Laplace transform of p(x, y, τ), moreover it can be inverted, see Craddock[Cra09] and Craddock and Lennox [CL09] for details, to have that the fundamental solution of (3.2)reads as follows

pCIR(x, y, τ) =2kek( 2kθ

σ2+1)τ

σ2(ekτ − 1)

(yx

) ν2

exp

(−2k(x+ ekτy)

σ2(ekτ − 1)

)Iν

(2k√xy

σ2 sinh(kτ2 )

), (3.4)

where ν := 2kθσ2 − 1. We have found a fundamental solution but we are not done yet, since this

does not guarantee us that this is a transition density function. Not even the condition, which is infact satisfied,

∫R p(x, y, τ)dy = 1, is sufficient to this intent. Using now proposition 2.2.5 we can

conclude that (3.28) is in fact the transition density function of the CIR process.Moreover, exploiting theorem 2.2.11, we can compute the ZCB fair price, see, e.g., Brigo and

Mercurio [BM06] for a description of such type of bond. In fact, supposing the ZCB associatedinterested rate is driven by the CIR process defined by equation (3.1), then its fair price is given bytheorem (2.2.11) setting

µ = 1 ; A = k2 + 2σ2 ; B = −k2θ , C =θk

2(θk − σ2) ;

F (x) = θk ln(x)− kx ; C1 = 1 ; C2 = 0

We recall the definition of fundamental solution for a parabolic operator. Given a differential


operator uτ − Lu = 0 with initial datum u(0, x) = f(x) for x ∈ Ω and τ ∈ [0, T ], a fundamentalsolution is a kernel p(x, y, τ) such that: (i) for fixed y, p(x, y, τ) is a solution of the PDE on Ω×(0, T ];(ii) u(x, τ) =

∫Ωf(x)p(x, y, τ)dy is a solution of the Cauchy problem for a given initial datum f .

We anyway refer to Friedman [Fri64] for a complete treatment of fundamental solutions for partialdifferential equations of parabolic type.

Therefore we can recover a classical solution from the fundamental solution using point (ii) ofthe previous definition, being Pt the price at time t of the ZCB. Setting

Pt = u(x, t) =

∫ ∞0

γx(σ2−2θk)/2σ2

y(σ2+2θk)/2σ2

ekσ2

(x−y)

σ2 sinh(γ(T − t)/2) ×

× exp

(k2θ(T − t)

σ2− γ(x+ y)

σ2 tanh(γ(T − t)/2)

)Iσ2−2kθ

σ2

(2γ√xy

σ2 sinh(γ(T − t)/2)

)dy

(3.5)

We can thus recover the following

u(x, t) = exp

[−2x

(eγ(T−t) − 1

)(γ + k)(eγ(T−t) − 1) + 2γ

] [2γ exp [(γ + k)(T − t)/2]

(γ + k)(eγ(T−t) − 1) + 2γ

] 2kθσ2

with γ =√k2 + 2σ2. As already said at the beginning here we have called the time variable

τ = T − t the time left to maturity merely for financial reason.

3.1.2 Zero-Coupon Bond Price

The Zero-Coupon bond price plays a fundamental role in financial mathematics. In fact havingan analytic formula for the fair price of a ZCB, we can recover the yield curve. Such a curve is ofparamount importance in pricing many financial instruments. Again we refer to Brigo and Mercurio[BM06] for an extensive treatment of the topic. We will now evaluate the Zero-Coupon bond pricewith short rate modelled by the CIR process using Lie’s symmetries.

Let us therefore suppose that the interest rate Rt is governed by the SDE

dXt = k(θ −Xt)dt+ σ√XtdWt (3.6)

We can say that the zero-coupon price with maturity T , recalling that the price at maturity isP (T, T ) = 1, has fair price given by

E

[φ(XT ) exp(−

∫ T

0

Xtdt)

]= E

[1 exp(−

∫ T

0

Xtdt)

]= P (0, T )

Via Feynman-Kac formula we have that it is solution of the parabolic problemut + σ2

2 xuxx + k(θ − x)ux − xu = 0

u(x, T ) = 1(3.7)


The standard approach is to guess a solution of the form

u(x, t) = a(t, T ) exp[−xb(t, T )]

in order to prove that the analytic solution of the PDE (3.7) is of the form

u(x, t) = exp

[−2x

(eγ(T−t) − 1

)(γ + k)(eγ(T−t) − 1) + 2γ

] [2γ exp [(γ + k)(T − t)/2]

(γ + k)(eγ(T−t) − 1) + 2γ

] 2kθσ2

(3.8)

We will used a different approach. Using Lie’s algebra we will find equation (3.8) as invariantsolution. What we will do in particular to determine a sub-algebra F of the Lie-algebra admitted bythe CIR equation that leaves the final condition and the boundary surfaces unchanged.

Proceeding as usual we find 7 determining equations given the infinitesimal generator of the form

v = ξ(x, t)∂x + τ(x, t)∂t + φ(x, u)∂u

ξu = 0 (3.9a)

τu = 0 (3.9b)

φuu = 0 (3.9c)

τx = 0 (3.9d)

−2xσ2φx,u + ξt − 2kxξx + 2θkξx + xσ2ξx,x + 2kxτt − 2θkτt + 2kξ = 0 (3.9e)

−2φt − 2xuφu + 2kxφx − 2kθφx,x + 2xuτt + 2xφ+ 2uξ = 0 (3.9f)

−2xξx + xτt + ξ = 0 (3.9g)

(3.9h)

Solving the previous system we obtain the coefficients (3.33)ξ(x, t) = e−γt(e2γtk1 + k2)x

τ(x, t) = eγtk1−e−γtk2+k3γγ

φ(x, t) = e−γt

σ2γ (k4eγtσ2γ + k2(θk(k − γ) + x(−γ + kγ)) + e2γtk1(−θγ(k + γ) + x(γ2 + kγ))) + eγtσγα))

(3.10)where ξ, τ and φ are the coefficients of the general infinitesimal generator

v = ξ(x, t)∂x + τ(x, t)∂t + uφ(x, t)∂u

and α(x, t) a general solution of equation (3.33).


The system (3.35) allows us to determine a basis for the Lie algebra to be

v1 = eγt(x∂x + 1

γ ∂t + ( θk(γ+k)σ2γ − (γ+k)

σ2 x)u∂u

)v2 = e−γt

(x∂x − 1

γ ∂t + ( θk(k−γ)σ2γ + (k−γ)

σ2 x)u∂u

)v3 = ∂t

v4 = u∂u

(3.11)

with γ =√k2 + σ2.

The commutator table is thus

v1 v2 v3 v4

v1 0 0 γv3 −γv4

v2 0 0 0 0v3 −γv3 0 0 2

γ v1 − 2θk2

γσ2 v2

v4 γv4 0 − 2γ v1 + 2θk2

γσ2 v2 0

We are looking for a subalgebra F that leaves invariant the boundary condition such as theboundary surfaces. Therefore we want an infinitesimal generator of the form

v =

4∑i=1

civi

such thatv[t− T ]|t=T = 0, v[u− 1]|u=1 = 0

In this case F is spanned by

v = (k[1− cosh(γ(t− T ))] + γ sinh(γ(t− T ))) ∂t+

− γ (k sinh(γ(t− T ))− γ cosh(γ(t− T )))x∂x+

+ 2 (kθ[1− cosh(γ(t− T ))] + γx sinh(γ(t− T )))u∂u

(3.12)

We have thus to look for an invariant solution of (3.7) that arise from (3.12).

Using the method of characteristic we have

dx

ξ=dt

τ=du

φ(3.13)

where as usual we have denoted by ξ, τ and φ the coefficients of ∂x, ∂t and ∂u in (3.12).

Integrating the first and second member in (3.13) gives one invariant

X1 =(γ(e2γ(T−t) − 1) + k(eγ(T−t) − 1)2

) eγtx

(3.14)


Integrating the second and third member in (3.13) gives instead

X2 = ueh(x,t)(

(γ + k)(e2γ(T−t) − 1) + 2γ) 2θkσ2 (3.15)

where

h(x, t) =2(e2γ(T−t) + 1)x

γ(e2γ(T−t) − 1) + k(eγ(T−t) − 1)2+θk(γ + k)t

σ2

Now we have that for an arbitrary function Ψ, X2 = Ψ(X1) is the most general invariant of (3.12).Therefore the corresponding group-invariant solution must have the form

u(x, t) = e−h(x,t)(

(γ + k)(e2γ(T−t) − 1) + 2γ)− 2θk

σ2 ×

×Ψ((γ(e2γ(T−t) − 1) + k(eγ(T−t) − 1)2

) eγtx

)

(3.16)

Substituting now (3.16) into (3.7) leads to

y4Ψ′′(y)− (Ay2 + By3)Ψ′(y)− (C + Dy)Ψ(y) = 0 (3.17)

with

A =4k2eγT

σ2, B =

2(θk − σ2)

σ2, C =

16e2γT (k2 + σ2

σ2, D =

8θkeγT

σ2

We can now make the change of variable

y =1

x, Ψ(y) = v(x)

followed by

x = zσ2

4k2 exp(γT ), v(x) = w(z)

ending up with the equation

zw′′ +

(2θk

σ2+ z

)w′ −

(σ2(k2 + σ2)

k4+

2θ

k

)w = 0 (3.18)

We can now group the terms depending on z in order to get[z

((d

dz+σ2 + k2

k2

)(d

dz− σ2

k2

))+

2θk

σ2

(d

dz− σ2

k2

)]w = 0 (3.19)

Introducing now the variable

ζ = (d

dz− σ2

k2)w (3.20)

we reduce equation (3.19) into

ζ ′ +

(σ2 + k2

k2+

2θk

σ2z

)ζ = 0 (3.21)


whose solution is

ζ = Cz−2θkσ2 exp

[−(

1 +σ2

k2

)z

](3.22)

with C a constant of integration. Combining now (3.20) and (3.22) we obtain the differential equation

w′ − σ2

k2w = ζ(z) (3.23)

Now the quadrature is easily done, and setting C = 0 for boundary condition we obtain

Ψ(y) = K exp

[4eγT

y

](3.24)

Now from (3.16) and (3.24) we have

u(x, t) = K exp

(−2(e2γ(T−t) + 1)x

γ(e2γ(T−t) − 1) + k(e2γ(T−t) − 1)2− θk(γ + k)t

σ2

)×

× ((γ + k)(eγ − 1) + 2γ)−2θk

σ2 exp

(4xeγ(T−t)

γ(e2γ(T−t) − 1) + k(e2γ(T−t) − 1)2

)=

= K exp

(−2x(e2γ(T−t) + 1)

(γ + k)(e2γ(T−t) − 1) + 2γ− θk(γ + k)t

σ2

)×

× ((γ + k)(eγ − 1) + 2γ)−2θk

σ2

(3.25)

Using now the final condition for the Zero-Coupon bond we find

K = 2γ2θkσ2 exp

(θk(γ + k)T

σ2

)such that (3.25) becomes

u(x, t) = exp

(−2x(e2γ(T−t) + 1)

(γ + k)(e2γ(T−t) − 1) + 2γ

)exp

((γ + k)(T − t)

2

2θk

σ2

)×

×(

2γ

(γ + k)(e2γ(T−t) − 1) + 2γ

) 2θkσ2

=

= exp

(−2x(e2γ(T−t) + 1)

(γ + k)(e2γ(T−t) − 1) + 2γ

)(2γ exp((γ + k)((T − t)/2))

(γ + k)(e2γ(T−t) − 1) + 2γ

) 2θkσ2

(3.26)

which is the analytic solution for the Zero-Coupon bond price under the CIR model.

3.2 The constant elasticity of variance (CEV) model

Empirical analysis of observed option prices, and observed phenomena like the volatility smile,led researchers to formulate option pricing models that included non-constant volatility. In responseto observations of an inverse relationship between share price and share price volatility, documentedby Fischer Black (1975), the Constant Elasticity of Variance (CEV) model was derived. Cox (1996)suggests that the CEV model is the simplest way to describe such an inverse relationship. In fact,

3.2 The constant elasticity of variance (CEV) model 97

the CEV model was derived after a direct request from Fischer Black to John Cox, for a share priceevolution model that includes an inverse dependence of volatility and the share price, as described inCox (1996).

The CEV process is a generalization of the CIR model introduced in 3.1, it will seen how in awhile, but whereas the CIR is widely used in modelling interest rate the CEV model appear dealingwith equity markets and in option pricing.

Let S = St : t ≥ 0 be the stock price governed by the CEV stochastic differential equationdSt = µStdt+ σSβ/2t dWt

S0 = s

with Wt a standard Wiener process, β < 2, β − 2 is the elasticity and σ > 0 a constant.The return variance κ(St, t) with respect to prices St has the following relationship

dκ(St, t)/dStκ(St, t)/St

= β − 2

it follows then integrating and exponentiating both side that

κ(St, t) = σ2Sβ−2t

If β = 2 then the elasticity is 0 and the prices are lognormally distributed and we recover thestandard Black&Scholes model.

If β = 1 we have the well known Cox-Ingersoll-Ross (CIR) model.We have to specify that studies on the CEV with β > 2 has been done by Emanuel & MacBeth.

3.2.1 CEV transition density function

Since exploiting the CEV transition density function is not an easy task, via Ito’s lemma, we willtransform the CEV into the CIR process. We will apply then the results we have already obtained andthen transform everything back to the CEV process in order to retrieve the density function of theCEV process.

Let us see how in details.Let us now define f(x) = x2−β . Therefore by I-D’s lemma we have

dXt =∂f

∂SdSt +

1

2σ2Sβt

∂2f

∂S2dt =

[(2− β)µStS

1−βt +

σ2Sβt2

(2− β)(1− β)S−βt

]dt+ σS

β2t (2− β)S1−β

t dWt =

=

[(2− β)µS2−β

t +σ2

2(2− β)(1− β)

]dt+ σ(2− β)S

1− β2t dWt =

=

[(2− β)µXt +

σ2

2(2− β)(1− β)

]dt+ σ(2− β)

√XtdWt


For the sake of simplicity we will now change the notation

k = (β − 2)µ

θ = σ2

2µ (β − 1)

σ = (2− β)σ

(3.27)

in order to recover the standard form of the CIR process studied in section 3.1.

We have therefore retrieved the process X = Xt : t ≥ 0 solution to the CIR SDEdXτ = k(θ −Xτ )dτ + σ√XτdWτ

X0 = x0

Applying now theorem 2.2.10 and using the inverse Laplace transform, see again 3.1 for details,we can recover the transition density for the CIR process.

pCIR(x, y, τ) =2kek( 2kθ

σ2+1)τ

σ2(ekτ − 1)

(yx

) ν2

exp

(−2k(x+ ekτy)

σ2(ekτ − 1)

)Iν

(2k√xy

σ2 sinh(kτ2 )

), (3.28)

whit ν = 2kθσ2 − 1 and

Iν(x) =x

2

ν∑n≥0

(x/2)2n

n!Γ(ν + n+ 1)

a modified Bessel function of the first kind of order ν.

Since our goal was to find and explicit formula for the CEV transition density, we have now touse again the transformation in order to recover the density of the original process.

Let us just recall first of all thatX = f(S) = S2−β and therefore S = f−1(X) = g(X) = X1

2−β .Since the function is clearly increasing, we know from probability 1.0.1 that this is enough to allowus to invert the transformation, we can easily evaluate the transition density to be

pCEV (x, y, t) =

∣∣∣∣∂f(y)

∂y

∣∣∣∣ pCIR(f(x), f(y), t)

Recalling that we originally renamed all the parameters, see (3.27), we can evaluate the probabilitytransition density for the CEV process as follows

pCEV (x, y, τ) =8µeµ(2β−3)t

σ2(e(β−2)µt − 1)

(√x

y

)(β − 1)2

(β − 2)

exp

−2µ(x2−β + e(β−2)µty2−β)

σ2(β − 2)(e(β−2)µt − 1)

I 1β−2

(2µ(xy)

2−β2

σ2(β − 2) sinh( (β−2)µt2 )

),

Lemma 3.2.1 (Feller,1951). Let p(x, y, t) the transition density function for x and t conditional to y


of a given process. Then an explicit fundamental solution of the parabolic problem

(P (x, t))t = (axP (x, t))xx − ((bx+ h)P (x, t))x , 0 < x <∞

is given by

p(x, y, t) =b

a(ebt − 1)

(e−btx

y

)h−a2a

exp

[−b(x+ yebt)

a(ebt − 1)

]I1− ba

(2b

a(1− ebt)(e−btxy)

12

)where Ik(x) is the modified Bessel function of the first kind of order k.

We can check that the transition probability density is the same as the one derived thoroughFeller’s lemma 3.2.1 with appropriate parameters. See Feller [Fel51] for details.

3.2.2 CEV option pricing model

Let now be S = St : t ≥ 0 the stock price driven by the CEV stochastic differential equationdSt = µStdt+ σSβ/2t dWt

S0 = s

Let furthermore Π be a portfolio and U(St) denote the value of an option in the portfolio Π and be ∆

the quantity of the stock in Π. That isΠ = U −∆S

An infinitesimal change in the portfolio in a time interval dt, setting ∆ = ∂U∂S (we chose ∆ in such a

way in order to get rid of the stochastic terms!), is given by

dΠ =

∂U

∂tdt+

∂U

∂SdS +

1

2

∂2U

∂S2dS2 − ∂U

∂SdS

=

(∂U

∂t+

1

2

∂2U

∂S2σ2Sβ

)dt

Since it does not appear any stochastic term the value of the portfolio is certain. Thus in order topreclude arbitrage the payoff must be equal to Πrdt, then(

∂U

∂t+

1

2

∂2U

∂S2σ2Sβ

)dt =

(U − ∂U

∂SS

)rdt

We thus have the following PDE

∂U

∂t+ sr

∂U

∂S+

1

2

∂2U

∂S2σ2Sβ − rU = 0 (3.29)

Let us now assume for instance that U(St, t) =: C(St, t) is a call option. Therefore we giveas boundary condition CT = max(ST −K, 0) with K the strike price. We will not go deeper intofinancial details. For any doubt on financial topics we refer to Shreve [Shr04] and Cvitanic andZapatero [CZ04].


We thus have to solve the following PDE∂C∂t + sr ∂C∂S + 1

2∂2C∂S2 σ

2Sβ − rC = 0

CT = max(ST −K, 0)(3.30)

The standard approach in literature is to rewrite everything in terms of the logarithm of the spotprice i.e. x = ln(S) and evaluating thus

∂C

∂S=∂C

∂x

1

S,

∂2C

∂S

2

=1

S2

∂2C

∂x2− 1

S

∂C

∂x

Equation (3.32) can thus be written as∂C∂t +

(r − 1

2σ2)∂C∂x + 1

2∂2C∂x2 σ

2ex(β−2) − rC = 0

CT = max(exT −K, 0)(3.31)

Then, in analogy with the Black&Scholes approach a solution of the form

Ct(x, t) = exP1(x, t)−Ke−r(T−t)P2(x, t)

is guessed. See Chuang, Hsu, and Lee [CHL10], Chen and Lee [CL10], and Hsu, Lin, and Lee[HLL08] for details.

We will now follow a different approach. As already done in section 3.2.1 we will transform theCEV process into the CIR process. Since it has done previously we will now avoid all the computationand we will start from the process S = Xt : t ≥ 0 satisfying the stochastic differential equationdXτ = k(θ −Xτ )dτ + σ

√XτdWτ

X0 = x0

We will consider again a portfolio Π composed as before, just this time the stock price is drivenby the CIR process.

We thus have the following

dΠ =

(∂U

∂t+σ2

2

∂2U

∂X2X

)dt =

(U − ∂U

∂XXr

)dt

Changing the variable t := T − t in order to have the forward Cauchy problem with initial valuemax(X0 −K), in analogy with (3.32) we get the following∂C

∂t = Xr ∂C∂X + σ2

2∂2C∂X2X − rC

C0 = max(X0 −K, 0)(3.32)


Ignoring for the moment the Cauchy problem we merely focus on the PDE

∂u

∂t= xr

∂u

∂x+σ2

2

∂2u

∂x2x− ru (3.33)

We would to compute the Lie algebra of equation (3.33). Proceeding as usual we can find sevendetermining equations given the general infinitesimal generator

v = ξ(x, t, u)∂

∂x+ τ(x, t, u)

∂

∂t+ φ(x, t, u)

∂

∂u

ξu = 0 (3.34a)

τu = 0 (3.34b)

φuu = 0 (3.34c)

τx = 0 (3.34d)

2xσ2φx,u + 2ξt − 2rxξx − xσ2ξx,x + 2rxτt + 2rξ = 0 (3.34e)

2φt − 2ruφu − 2rxφx − xσ2φx,x + 2ruτt + 2rφ = 0 (3.34f)

−2xξx + xτt + ξ = 0 (3.34g)

(3.34h)

The previous system leads to the most general form of the coefficients of the infinitesimalsymmetry of equation (3.33)

ξ(x, t) = e−rt(e2rtk1 + k2)x

τ(x, t) = ertk1−e−rtk2+k3rr

φ(x, t) = e−rtk2 + k4 − ertk1(2rx+σ2)σ2 + α(x, t)

(3.35)

where ξ, τ and φ are the coefficients of the general infinitesimal generator

v = ξ(x, t)∂x + τ(x, t)∂t + uφ(x, t)∂u

and α(x, t) a general solution of equation (3.33).

The system (3.35) allows us to determine a basis for the Lie algebra to be

v1 = ertx∂x + ert

r ∂t − ( 2rx+σ2

σ2 )ertu∂u

v2 = e−rtx∂x − e−rt

r ∂t + e−rtu∂u

v3 = ∂t

v4 = u∂u

(3.36)

The commutator table is thus


v1 v2 v3 v4

v1 0 − 2v3r + 2v4 rv1 0

v22v3r − 2v4 0 −rv2 0

v3 −rv1 rv2 0 0v4 0 0 0 0

We are looking for those generators v who preserves the boundary conditions. Therefore werequire

v[x]|x=0 = 0, v[t]|t=0 = 0, v[x−K]|x=K = 0

Those invariance lead us to the following systemτ(0) = 0⇒ k1 − k2 + k3r = 0

ξ(K) = 0⇒ ertKk1 + e−rtKk2 = 0(3.37)

Since the last condition holds for any t the following easily follows Those invariance lead us to thefollowing system

τ(0) = 0⇒ k1 − k2 + k3r = 0

k1 = −k2

k3 = 2k2r

(3.38)

We can now reformulate the system (3.35) asξ(x, t) = k2(e−rt − ert)x

τ(x, t) = k2r (2− e−rt − ert)

φ(x, t) = k4 + k2(e−rt + ert( 2rx+σ2

σ2 ))

(3.39)

We would like to recover a fundamental solution. Thus we prescribe u(x, 0) = δ(x− y) as initialdatum, with δ the Dirac mass. In order to such a condition to hold we require

φ(x, 0)δ(x− y) = ξ(x, 0)δ′(x− y)

Thus we get the following ξ(y, 0) = 0

φ(y, 0) = − ∂ξ∂x (y, 0)

Substituting now the last equation into (3.39) we get

φ(y, 0) = 0⇒ k4 = −k2(2σ + 2ry

σ2)


Therefore we get the following coefficientsξ(x, t) = k2(e−rt − ert)x

τ(x, t) = k2r (2− e−rt − ert)

φ(x, t) = k2

(e−rt + ert( 2rx+σ2

σ2 )− 2σ2+2ryσ2

) (3.40)

We then compute the invariants for the action of

v = ξ∂

∂x+ τ

∂

∂t+ uφ

∂

∂u(3.41)

with ξ, τ and φ given as in (3.40).

Equivalently

v = k2

(−v1 + v2 +

2

rv3 − 2

σ2 + yr

σ2v4

)(3.42)

or in explicit form

v =k2[(e−rt − ert)x∂x +

((2− e−rt − ert)

r

)∂t+

+

(e−rt + ert

(2rx+ σ2

σ2

)−(

2σ2 + 2ry

σ2

))u∂u]

(3.43)

Invariance of the action are given by solving the equation v[η] = 0.

We can find invariance by solving the characteristic system

dx

ξ=dt

τ=du

φ(3.44)

Integrating the first and second part of equation (3.44) we get the invariant

η = η(x, t) =e−rt(1− ert)2

x=

2(cosh(rt)− 1)

x(3.45)

In order to find the other invariant we replace

x =2(cosh(rt)− 1)

η

noting that η is constant for all solutions of the characteristic system. Thus we get the second invariant

v = exph(x, t)u(x, t) = expr[−2y + tσ2 + 2ert(y − tσ2) + e2rt(2x+ tσ2)

]σ2(ert − 1)2

u(x, t)

Thus since η is a basis for the invariants it follows that v = Ψ(η) for a generic function Ψ. Thus wehave the following general form of a solution u(x, t)

u(x, t) = exp−h(x, t)Ψ(η) (3.46)


Substituting now equation (3.46) into equation (3.33) we get that Ψ(η) has to satisfy

−r2x3 (x+ 2y − 2y cosh(rt))Ψ(η) + 16xσ2[rx− σ2 + σ2 cosh(rt)

]sinh4(

rt

2)Ψ(η)′+

+ 64σ4 sinh8(rt

2)Ψ′′(η) = 0

(3.47)

3.2.3 Mapping

Recalling theorem 1.4.3 we can see that since the Lie algebra of equation (3.33) is spanned byfour vector fields we can find a one-to-one mapping to the partial differential equation

vy = vττ −A

y2v (3.48)

whose Lie algebra is spanned by

w1 = τy∂y + τ2∂τ −(

14y

2 + 12τ)v∂v

w2 = ∂τ

w3 = y∂y + 2τ∂τ

w4 = v∂v

(3.49)

with commutator table

w1 w2 w3 w4

w1 0 −w3 + 12 w4 −2w1 0

w2 w3 − 12 w4 0 2w2 0

w3 2w1 −2w2 0 0w4 0 0 0 0

We recall that the backward parabolic equation

ut = xrux +σ2

2xuxx − ru (3.50)

has a four dimensional Lie algebra spanned by

v1 = ertx∂x + ert

r ∂t − ( 2rx+σ2

σ2 )ertu∂u

v2 = e−rtx∂x − e−rt

r ∂t + e−rtu∂u

v3 = ∂t

v4 = u∂u

(3.51)

and commutator table


v1 v2 v3 v4

v1 0 − 2v3r + 2v4 rv1 0

v22v3r − 2v4 0 −rv2 0

v3 −rv1 rv2 0 0v4 0 0 0 0

If we consider the Lie subalgebra h(23) spanned by v2 and v3 (equivalently by w1 and w3) andconsidering the scaling wi = αiwi with i = 2, 3, the the commutators arising from wii are thesame as the commutators in table (3.2.3) arising from vii if

α2 = − 8r

σ2, α3 =

2

r(3.52)

This suggests seeking a one-to-one point transformation mapping of the PDE (??) to the targetPDE (3.48) such that it maps each vi to the corresponding wi with

w2 = − 8r

σ2w2, w3 =

2

rw3 (3.53)

In particular we look for a transformation of the formy = α(x, t)

τ = β(t)

v = γ(x, t, u)

(3.54)

Such a transformation has to satisfy conditions

vjα(x, t) = wjy|(y,τ,v)=(α,β,γ), vjβ(t) = wjτ |(y,τ,v)=(α,β,γ)

vjγ(x, t, u) = wjv|(y,τ,v)=(α,β,γ)

(3.55)

with j = 2, 3 System (3.55) read extensively

e−rtxαx +e+rt

rαt = 0 (3.56a)

αt =r

2α (3.56b)

−e−rt

rβt = −σ

2

8r(3.56c)

βt = rβ (3.56d)

e−rtxuγx −e−rt

ruγt + e−rtuγ = 0 (3.56e)

uγt = 0 (3.56f)

(3.56g)


Solutions of the previous system arey = α(x, t) = exp rt2

√x

τ = β(t) = −σ2

8r exprt

v = ux

(3.57)

We would like to stress that all the constants was chosen appropriately after careful computation.

Since estimation are tough to perform we prefer to proceed step by step. We first of all apply thefollowing change of variables

x = y2σ2

8rτ

t =log( 8rτ

σ2)

r

u(x, t) = v(y(x, t), τ(x, t)) = v(

exp rt2 √x, exprtσ

2

8r

) (3.58)

By the standard chain rule we can express the partial derivatives of u w.r.t. variables (x, t) in termof partial derivative of v w.r.t. variables (y, τ) as

ut =√x r2 exp rt2 vy + σ2

8 exprtvτux = 1

2√x

exp rt2 vyuxx = 1

4x exprtvyy − 1

4x32

exp rt2 vy

(3.59)

Substituting now equation (3.59) into equation (3.50) we get the following

vτ = vyy −4

yvy −

1

τv (3.60)

Applying now the change of variable

v(y, τ) =y2σ2

8rτv(y, τ) (3.61)

and computing the partial derivatives we getvτ = − 1

τ2y2σ2

8r v + y2σ2

8rτ vτ

vy = 2yσ2

8rτ v + y2σ2

8rτ vy

vyy = 2σ2

8rτ v + 4yσ2

8rτ vy + y2σ2

8rτ vyy

(3.62)

We thus get the equation

vτ = vyy −6

y2v (3.63)

This equation has a fundamental solution of the form

p(y, y, τ) =

√yy

2τexp−y

2 + y2

4τI 5

2

(yy

2τ

)(3.64)

3.3 The Stochastic-alpha-beta-rho (SABR) model 107

3.3 The Stochastic-alpha-beta-rho (SABR) model

In this section we will study a model where the volatility of stocks return is assumed to bestochastic. We will give particular emphasis to the model proposed by Heston. For some furthernotions on stochastic volatility model we refer to Brigo and Mercurio [BM06] and Joshi [Jos08].

The SABR model is a stochastic volatility model which attempts to capture the volatility smile inderivatives markets. The CEV, studied in section ?? is an important ingredient of such a model.

Let us assume that the stock price S = St : t ≥ 0 is driven by the process under the risk neutralmeasure

dSt = rStdt+ ν(t)β2 StdW

1t (3.65)

where µ(t) is the instantaneous drift of stock return, νt the variance and W 1t a standard Wiener

process. We assume further that the variance V = Vt : t ≥ 0 follows

νt = µν(S, ν, t)dt+ σνναβ(S, ν, t)dW 2

t (3.66)

where W 2t is again a standard Wiener process with dW 1

t dW2t = ρdt, −1 ≤ ρ ≤ 1, η is the volatility

of the volatility and ρ is the correlation between the two Wiener processes. It is often consideredβ = 1 in equation (3.65) and α = 1

2 , anyway we can always pass from a process to the other via Ito’slemma. In the proceeding we will always consider β = 1 and α = 1

2 .

Lemma 3.3.1 (Ito’s lemma). Let f be a function with independent variables (x, t) ∈ RN × R+. Let

us suppose that Xi follows the Ito process

dXit = ai(x, t)dt+ bi(x, t)dW

it

where as usual the W it are standard Wiener processes with correlation coefficient dW idW j = ρijdt.

Then

df =

∑i

∂f

∂xiai +

∂f

∂t+

1

2

∑ij

∂2f

∂xi∂xjbibjρij

dt+∑i

∂f

∂xibidW

it (3.67)

Since we are dealing with two sources of randomness we have to hendge both the stock price andthe volatility. Thus we can set up a portfolio Π with the option we are to price, denoted by U(S, ν, t),a quantity −∆ of the stock plus a quantity −∆1 of another asset with value U1(S, ν, t). Thereforewe have

Π = U(S, ν, t)−∆S −∆1U1(S, ν, t)

Applying now lemma 3.3.1 with N = 2 we get

dU =∂U

∂SdS +

∂U

∂νdν +

(∂U

∂t+

1

2νS2 ∂

2U

∂S2+ ρηνβS

∂2U

∂ν∂S+

1

2η2νβ2 ∂

2U

∂ν2

)dt

dU1 =∂U1

∂SdS +

∂U1

∂νdν +

(∂U1

∂t+

1

2νS2 ∂

2U1

∂S2+ ρηνβS

∂2U1

∂ν∂S+

1

2η2νβ2 ∂

2U1

∂ν2

)dt


dΠ = dU −∆dS −∆1dU1 =

=

(∂U

∂t+ν

2S2 ∂

2U

∂S2+ ρηνβS

∂2U

∂S∂ν+

1

2η2νβ2 ∂

2U

∂ν2

)dt

−∆1

(∂U1

∂t+ν

2S2 ∂

2U1

∂S2+ ρηνβS

∂2U1

∂S∂ν+

1

2η2νβ2 ∂

2U1

∂ν2

)dt(

∂U

∂S−∆1

∂U1

∂S−∆

)dS(

∂U

∂ν−∆1

∂U1

∂ν

)dν

(3.68)

Setting now∂U

∂S−∆1

∂U1

∂S−∆ = 0

and∂U

∂ν−∆1

∂U1

∂ν= 0

we can eliminate all randomness from the portfolio. Thus from arbitrage assumption we get

dΠ = dU −∆dS −∆1dU1 =

=

(∂U

∂t+ν

2S2 ∂

2U

∂S2+ ρηνβS

∂2U

∂S∂ν+

1

2η2νβ2 ∂

2U

∂ν2

)dt

−∆1

(∂U1

∂t+ν

2S2 ∂

2U1

∂S2+ ρηνβS

∂2U1

∂S∂ν+

1

2η2νβ2 ∂

2U1

∂ν2

)dt =

= rΠdt = r(U −∆S −∆1U1)dt

(3.69)

We can collect now all U terms on the left side and all the U1 terms on the right side ending upwith

∂U∂t + ν

2S2 ∂2U∂S2 + ρηνβS ∂2U

∂S∂ν + 12η

2νβ2 ∂2U∂ν2 + rs∂U∂S − rU

∂U∂ν

=

=∂U1

∂t + ν2S

2 ∂2U1

∂S2 + ρηνβS ∂2U1

∂S∂ν + 12η

2νβ2 ∂2U1

∂ν2 + rS ∂U1

∂S − rU1

∂U1

∂ν

(3.70)

Since the left hand side of equation (3.70) is a function of U alone and the right hand side dependson U1. The only way this can be true is that both terms are equal to a function f(S, ν, t).

3.3.1 The Heston model

The Heston model has been proposed by Heston in 1993 in Heston [Hes93]. It is a particular caseof the general model introduced in the previous section. In particular it correspond to the case of

µν(s, ν, t) = κ(θ − νt), β = 1, α =1

2


in equation (3.66). Therefore we are dealing withdSt = rSdt+√νtStdW

1t

dνt = κ(θ − νt)dt+ σν√νtdW

2t

(3.71)

with dW 1dW 2 = ρdt, κ the speed of reversion of νt and θ the long-term mean of volatility process.

Recalling now what we said at the end of the previous section we get

∂U

∂t+ν

2S2 ∂

2U

∂S2+ ρσνS

∂2U

∂S∂ν+

1

2σ2ν

∂2U

∂ν2+ rs

∂U

∂S− rU + (κ(θ − νt)− λ(S, ν, t))

∂U

∂ν= 0

(3.72)

If we consider the case of a European call option we have the following boundary condition

U(S, ν, T ) = max(ST −K, 0)

U(0, ν, t) = 0

∂U∂S (∞, ν, T ) = 1

rS ∂U∂S (S, 0, t) + κθ ∂U∂ν (S, 0, t)− rU(S, 0, t) + ∂U∂t (S, 0, t) = 0

U(S,∞, t) = S

(3.73)

Usuallyin analogy to the Black&Scholes model a solution of the form

U(S, ν, t) = SP1 −Ke−r(T−t)P2

is guessed. As done in section ?? we will have a non standard approach using Lie’s algebras.

We will consider first of all the much simpler case of uncorrelated Wiener processes. Then wehave to perform some change of variables in order to consider a easier parabolic problem.

Let us in fact perform a change of variable in order to have a well-posed forward parabolicproblem and setting furthermore

t := T − t, X = ln(S), U(t, S, ν) = u(t, ln(S), ν)

by the chain rule we have US = uxxS = ux

1S

USS = 1S2uxx − 1

S2ux

Ut = −ut

(3.74)

substituting now into equation (3.72) we get the following problem

ut =ν

2uxx + (r − ν

2)ux +

σ2

2νuνν + κ(θ − νt)uν − ru (3.75)


We apply now the Fourier transform to the function u(t, x, ν, t) in the variable x

u(t, ξ, ν) := F [u(t, x, ν)] (ξ) =

∫Ru(t, x, ν)e−iξxdx

Recalling now the usual differentiation rule for the Fourier transform

ˆDαu = i|α|ξαu

we have that the only derivatives that change areD2xu = −ξ2u

Dxu = iξu(3.76)

Thus we get the following parabolic problem

ut =σ2

2νuνν + κ(θ − νt)uν − νµu+ αu (3.77)

Setting now V = e−αtu we get eventually the following PDE

Vt =1

2σ2νVνν + κ(θ − νt)Vν − µνV (3.78)

withµ =

1

2(ξ2 − iξ), α = r(iξ − 1)

We would like to recover a fundamental solution p(x, y, t) of equation (3.78). In fact given such afundamental solution we have that

U(ξ, ν, t) = eα(T−t)∫φ(ξ, y)p(T − t, ν, y)dy (3.79)

with φ the Fourier transform of an appropriate initial datum φ.

We can now see that equation (3.78) is of the form (2.28). Applying thus theorem 2.2.11 we canobtain a fundamental solution of the form (2.29)

p(ν, y, t) =

√Aνye−(F (ν)−F (y)/(σ2))

σ2 sinh(√At/2)

exp

(−Btσ2−

√A(ν + y)

σ2 tanh(√At/2)

)(C1(y)In

(2√Aνy

σ2 sinh(√At/2)

)+ C2(y)I−n

( √Aνy

σ sinh(√At/2)

)) (3.80)

with n = 2√σ4+8C√2σ2

, the constants A, B and C such that f = κ(θ− νt) solution of the second Riccatiequation

σνf ′ − σf +1

2f2 + 2σµν2 =

1

2Aν2 +Bν + C, with A > 0 (3.81)

and F ′(x) = f(x)x .


Thus given an appropriate initial datum U(0, S, ν) = φ(S, ν) the evaluation of equation∂U∂t + ν

2S2 ∂2U∂S2 + 1

2σ2ν ∂

2U∂ν2 + rs∂U∂S − rU + f(ν)∂U∂ν = 0

U(0, S, ν) = φ(S, ν)(3.82)

is reduced to the evaluation of the integral (3.79)

U(ξ, ν, t) = eα(T−t)∫φ(ξ, y)p(T − t, ν, y)dy

with p(T − t, ν, y) of the form (3.80).We thus get

κ+ 2σ2µ = A, B = −κ2θ, C =σ2

2κθ +

κ2

2θ2, F (x) = κθ ln(x)− κx

Appendix A

Appendix

A.1 Complements on Lie group

A.1.1 Orbits

We recall the definition of G-orbit

Definition A.1.1 (G-orbit). A G-orbit of a local transformation group is a minimal non empty group

invariant subset of the manifold M.

A set O ⊂M is an orbit if it satisfies:

(i) if x ∈ O, ε ∈ G and exp(εv)(x) is defined then x = exp(εv)(x) ∈ O;

(ii) if O ⊂ O and O satisfies part (i) then either O = O or O is empty.

If we are dealing with a global transformation group, for each x ∈ M the orbit through x isdefined as follows

Ox := exp(εv)(x) : ε ∈ G

Definition A.1.2. Let G a local group of transformations action on a manifold M.

1. The group G acts semi-regularly if all the orbits O are as the same dimension as submanifolds

of M;

2. The group G act regularly if the action is semi-regular and in addition for each x ∈M it exists

a neighbourhood U of x with the property that each orbits of G intersects U in a pathwise

connected subset.

A.1.2 The differentials

Let be M and N two manifolds. A smooth map F : M → N between manifolds will mapsmooth curves on M to smooth curves on N inducing thus a map between their tangent vectors. Theresult is a linear map dF : TM |x → TN |F (x) between the tangent spaces of the two manifolds,

113

114 A. Appendix

called the differential of F. To be more rigorous, if the parametrized curve φ(t) has tangent vectorv|x = φ′(t) at the point x = φ(t), then the image curve ψ(t) = F (φ(t)) will have tangent vectorw|y = dF ()v|x = ψ′(t) at the image point y = F (x). Alternatively, if we treat tangent vectors asderivations, then we can define the differential by the chain rule formula

dF (v|x)[h(y)] = v[h F (x)], for any h : N→ R

In terms of local coordinates

dF (v|x) = dF

(m∑i=1

ξi∂

∂xi

)=

n∑j=1

(m∑i=1

ξi∂f j

∂xi

)∂

∂yj(A.1)

An important remark is that, in general, unless F is one-to-one, its differential dF does not mapvector fields to vector fields. Indeed if v is a vector field on M and x and x are two points in M withthe same image F (x) = F (x) in N , there is no reason why dF (v|x) should necessarily agree withdF (v|x). However, if v is mapped to a well-defined vector field dF (v) on N , then the two flowsmatch up, meaning

F [exp(εv)x] = exp(εdF (v))F (x)

Moreover, the differential dF respects the Lie bracket operation

dF ([v,w]) = [dF (v), dF (w)]

whenever dF (v) and dF (w) are well-defined vector fields on N .

Example A.1.1. Let M = R2 with coordinates (x, y) and let N = R with coordinate s, and letF : R2 → R be any map s = F (x, y). Given

v|(x,y) = a∂

∂x+ b

∂

∂y

then by (A.1)

dF (v|(x,y)) =

a∂F

∂x(x, y) + b

∂F

∂y(x, y)

d

ds|F (x,y)

A.1.3 Lie Algebras

The Lie algebra can be treated more extensively considering left or right multiplication. We preferto leave this part as complementary since it is more of an algebraic treatment of the Lie group. For areader interested more in the application this part can be skipped without any consequence but it is ofgreat importance for the foundational of the theory.

The most important example of Lie algebra is provided by the action of a Lie group G on itself byleft or right multiplication. Here, the invariant vector fields determine the Lie algebra or infinitesimalLie group. Given ε ∈ G, we let Lε : h 7→ ε h and Rε : h 7→ h ε denote the associated left and rightmultiplication maps. A vector field v on G is called left-invariant if dLε(v) = v, and right-invariant if

A.1 Complements on Lie group 115

dRε(v) = v, for all ε ∈ G.

Definition A.1.3. The left (respectively right) Lie algebra of a Lie group G is the space of all

left-invariant (respectively right-invariant) vector fields on G.

As said to any Lie group there are associated two different algebras: gL and gR. We have assumedwe always refer to the right algebra. Anyway the reader should know a left algebra can be constructedas well.

Every right-invariant vector field v is uniquely determined by its value at the identity e, becausev|ε = dRε(v|e). Thus we can identify the right Lie algebra with the tangent space to G at the identityelement, g ' TG|e, so g is a finite-dimensional vector space having the same dimension as G.

Each Lie algebra associated with a Lie group comes equipped with a natural bracket, inducedby the Lie bracket of vector fields. This follows immediately from the invariance of the Lie bracketunder diffeomorphisms, which implies that if both v and w are right-invariant vector fields, so is theirLie bracket [v,w].

Definition A.1.4 (Lie-algebra). A Lie algebra g is a vector space over some field F with an additional

law of combination of elements (the Lie bracket)

[·, ·] : g× g→ g

such that ∀a, b ∈ F and ∀v1, v2, v3 ∈ g the following properties hold:

(Closure) [vi, vj ] ∈ g;

(Bilinearity) [vi, avj + bvk] = a[vi, vj ] + b[vi, vk];

(Anticommutativity) [vi, vj ] = −[vj , vi];

(Jacobi identity) [vi, [vj , vk]] + [vj , [vk, vi]][vk, [vj , vi]]=0;

for any vi, vj , vk ∈ g.

Example A.1.2. The Lie algebra gl(n) of the general linear group GL(n) can be identified withthe space of all n× n matrices. In terms of the coordinates provided by the matrix entries

X = (xij) ∈ GL(n)

the left-invariant vector field associated with a matrix

A = (aij) ∈ gl(n)

116 A. Appendix

has the explicit formula

vA =

n∑i,j,k=1

xikakj∂

∂xij(A.2)

The Lie bracket of two such vector fields is

[vA, vB ] = vC

where C = AB −BA, so the left-invariant Lie bracket on GL(n) can be identified with the standardmatrix commutator

[A,B] = AB −BA.

On the other hand, the right-invariant vector field associated with a matrix A ∈ gl(n) is given by

va =

n∑i,j,k=1

xikakj∂

∂xij(A.3)

Now the Lie bracket is[vA, vB ] = vC

where C = −C = BA− AB is the negative of the matrix commutator. Thus, the matrix formulafor the Lie algebra bracket on gl(n) depends on whether we are dealing with its left-invariant orright-invariant version.

A.1.4 The exponential map

From a philosophical perspective, the exponential function may be viewed as a

link between the seemingly contradictory positions of Heraclitus on the one side and

Parmenides on the other. While the time-dependent function t→ T (t) reflects the aspect

of permanent change in a deterministic autonomous system, its generator A stands for

the eternal, timeless principle behind the system. The exponential function ties both

aspects together.1

K.J. Engel, R. Nagel

Given a right-invariant vector field v ∈ g on the Lie group G, we let

exp(εv) : G→ G

denote the associated flow. Applying the flow to the identity element e serves to define the one-

parameter subgroup

exp(εv) ≡ exp(εv)e

the vector field v is known as the infinitesimal generator of the subgroup. The notation is notambiguous, since the flow through any g ∈ G is the same as left multiplication by the elements of the

1The author may wish to thank S.B. for the hint.

A.1 Complements on Lie group 117

subgroup, so exp(εv)g can be interpreted either as a flow or as a group multiplication. Vice versa,the infinitesimal generators of the action of G on itself by right multiplication is the Lie algebragL left-invariant vector fields. This interchange of the role of infinitesimal generators and invariantvector fields is one of the interesting peculiarities of Lie group theory. Finally, note that although theleft- and right-invariant vector fields associated with a given tangent vector v ∈ TG|e are (usually)different, and have different flows, nevertheless the associated one-parameter groups coincide

exp(εvL) = exp(εvR)

Example A.1.3. Let us consider again the general linear group GL(n) previously discussed.

The flow corresponding to the right-invariant vector field vA (A.3) is given by left multiplicationby the usual matrix exponential

exp(εvA)X = eεAX

Conversely, the flow corresponding to the left-invariant vector field vA given by (A.2) is given byright multiplication

exp(εvA)X = XeεA

In either case, the one-parameter subgroup generated by the vector field associated with a matrixA ∈ gl(n) is the matrix exponential

exp(εvA) = exp(εvA) = eεA

exp(tvA) = exp(tbvA) = etA.

From now on we shall restrict our attention to the right-invariant vector fields, so that g willalways denote the right Lie algebra of the Lie group G. Evaluation of the flow exp(εv) at ε = 1 foreach v ∈ g serves to define the exponential map exp : g→ G. Since exp(0) = e, d exp(0) = 1, theexponential map defines a local diffeomorphism in a neighbourhood of 0 ∈ g. Consequently, all Liegroups having the same Lie algebra look locally the same in a neighbourhood of the identity; onlytheir global topological properties are different.

A.1.5 Infinitesimal groups actions

Just as a one-parameter group of transformations is generated as the flow of a vector field, so ageneral Lie group of transformations G acting on a manifold M will be generated by a set of vectorfields on M , known as the infinitesimal generators of the group action.

Each infinitesimal generators flow coincides with the action of the corresponding one-parametersubgroup of G. Specifically, let us suppose G is a local Lie group of transformations acting on amanifold M via exp(εv)(x). Then if v ∈ g we define ψ(v) to be the vector field on M whose flowcoincides with the action of the one-parameter subgroup exp(εv) of G on M .

Thusψ(v)|x =

d

dε|ε=0Ψ(exp(εv)(x) = dΨv|e(x) (A.4)

118 A. Appendix

Note now thatΨ(x) Rε(δ) = Ψεδ(x) = Ψδ(Ψε(x))

Thus we havedΨv|ε(x) = dΨv|e(Ψε(x)) = ψ(v)|Ψε(x)

It follows immediately from the properties of the Lie bracket that

[ψ(vi), ψ(vj)] = ψ([vi, vj ])

Theorem A.1.5. Let us consider wjj=1,...,r vector field on a manifold M satisfying

[wi,wj ] =

r∑k=1

Ckijwk, i, j = 1, . . . , r

for certain constants Ckij . Then there is a Lie group G whose Lie algebra has the given Ckij as

structure constants relative to some basis vjj=1,...,r and a local group action of G on M such that

ψ(vi) = wi where ψ is defined as in (A.4).

The differential dΨx preserves the Lie bracket between vector fields; therefore the resultingvector fields form a finite-dimensional Lie algebra of vector fields on the manifold M , satisfying thesame commutation relations as the right Lie algebra g of G.

Just as every Lie algebra generates a corresponding Lie group, given a finite-dimensional Lie alge-bra of vector fields on a manifold M , we can always reconstruct a (local) action of the correspondingLie group via the exponentiation process.

Theorem A.1.6. Let g be a finite-dimensional Lie algebra of vector fields on a manifold M . Let

G denote a Lie group having Lie algebra g. Then there is a local action of G whose infinitesimal

generators coincide with the given Lie algebra.

Appendix B

Appendix

B.1 Fundamental solutions for parabolic equations

Ah, Avner Avner...il buon vecchio Avner.

Ah, Avner Avner...good old Avner.

Anonymous

We are here to give some general notions on linear second-order parabolic equations and funda-mental solutions. For a deeper treatment see Evans [Eva10] and Friedman [Fri64].

Second-order parabolic equations are natural generalizations of the heat equation seen in section??.

B.1.1 Definitions

We will assume further U to be an open, bounded subset of RN and we set UT = U×]0, T ] forsome fixed T > 0.

We will then study the Cauchy-Dirichlet problem (CD problem)ut + Lu = f, in UT

u = 0, on ∂U × [0, T ]

u = g, on U × t = 0

(B.1)

where f : UT → R and g : U → R are given functions and u : UT → R is the unknown, u = u(x, t)

with x ∈ RN and t ∈ [0, T ]. For each time t L denotes a second-order linear differential operatoreither in the diverge form

Lu = −N∑

i,j=1

(aij(x, t)uxi

)xj

+

N∑i=1

bi(x, t)uxi + c(x, t)u (B.2)

119

120 B. Appendix

or the non-divergence form

Lu = −N∑

i,j=1

aij(x, t)uxixj +

N∑i=1

bi(x, t)uxi + c(x, t)u (B.3)

both for given coefficients ai,j , bi and c.

Definition B.1.1. We say that the partial differential operator ∂t +L is (uniformly) parabolic if there

exists a constant K > 0 s.t.N∑

i,j=1

ai,j(x, t)ξiξj ≥ K|ξ|2

∀ (x, t) ∈ UT , ξ ∈ RN .

Remark B.1.2. Let us notice that for each fixed time 0 ≤ t ≤ T the operator is uniformly elliptic in

the spacial variable x.

Remark B.1.3. If we take aij ≡ δi,j , bi ≡ c ≡ f ≡ 0 such that L = −∆, where we have denoted

by δi,j the Kronecker delta and ∆ =∑Ni=1

∂2

∂x2i

the Laplacian operator, the PDE becomes the

well-known heat equation.

In general a second order parabolic operator describes the time-evolution of the density of somequantity u. In our application u will mostly represents the payoff function of some particular financialinstrument.

B.1.2 The fundamental solution of the heat equation

We will now focus on the N -dimensional heat equationut −∆u = 0, in RN×]0,∞]

u = g, in RN × t = 0(B.4)

In this section it will come to light why we deal with fundamental solutions and in particular theyplay such an important role in studying parabolic equations.

To help intuition, think for instance of this special solution as the concentration of a substance oftotal mass q and suppose we want to keep the total mass equal to q at any time.

We observe that the heat equation involves one derivative with respect to the time variable tbut two derivatives with respect to the space variable x. Consequently if u solves (B.4) then sodoes u(λx, λ2t) for λ ∈ R. This scaling indicates the ratio |x|

2

t plays an important role for the heatequation (the astute reader will notice that we have already meet this symmetry in section ??).

We will therefore look for a solution of (B.4) having the form u(x, t) = v( |x|2

t ). Although thisapproach eventually leads to what we want, it is quicker to seek a solution u having the special form

u(x, t) =1

tαv( xtβ

)(B.5)

B.1 Fundamental solutions for parabolic equations 121

where α, β and v are to be found. We are here asking

u(x, t) = λαu(λβx, λt)

Inserting now (B.5) into (B.4) we obtain

αt−(α+1)v(y) + βt−(α+1)y ·Dv(y) + t−(α+2β)∆v(y) = 0 (B.6)

for y := t−βx. In order now to transform (B.6) into an expression involving the variable y alone wetake β = 1

2 . Therefore we have

αv +1

2y ·Dv + ∆v = 0 (B.7)

If we guess further that v is radial, i.e. v(y) = w(|y|) (B.7) becomes

αw +1

2rw′ + w′ + w′′ +

n− 1

rw′ = 0

for r = |y|. If we now set α = n2 we obtain

(rn−1w′

)′ 1

2(rnw)

′= 0

Thusrn−1w′ +

1

2rnw = a

Assuming now limr→∞ w,w′ = 0 we can conclude a = 0. Whence

w′ = −1

2rw

Then for some constant bw = be−

r2

4 (B.8)

Combining now (B.5) and (B.8) and our choice of α and β we conclude that

b

tp2

e|x|24t (B.9)

solves the heat equation (B.4).

Definition B.1.4. The function

Φ(x, t) :=

1

(4πt)p2e|x|24t , (x ∈ RN , t > 0)

0, (x ∈ RN , t < 0)(B.10)

is called the fundamental solution for the heat equation.

The choice of the normalizing constant (4π)−n/2 is dictated by the following

122 B. Appendix

Lemma B.1.5 (Integral of fundamental solution). For each time t > 0∫RN

Φ(x, t)dx = 1

We want now to use Φ to recover a solution to the C-D problem (B.4). Let us notice that thefunction (x, t) 7→ Φ(x, t) solves the heat equation away from the singularity at (0, 0) and so does(x, t) 7→ Φ(x− y, t) for each fixed y ∈ RN . Consequently the convolution

u(x, t) =

∫RN

Φ(x− y, t)g(y)dy (B.11)

=1

(4πt)p/2

∫RN

e−|x−y|2

4t g(y)dy (B.12)

should also be a solution.

Theorem B.1.6 (Solution of Cauchy-problem). Assume g ∈ C(RN ) ∩ L∞(RN ) and define u by

(B.11). Then

(i) u ∈ C∞(RN×]0,∞];

(ii) ut(x, t)−∆u(x, t) = 0;

(iii) lim(x,t)→(x0,0)u(x, t) = g(x0) for each xo ∈ RN .

Remark B.1.7. In view of Theorem B.1.6 we will writeΦt −∆Φ = 0, in RN×]0,∞[

Φ = δ0, on RN × t = 0(B.13)

δ0 denoting the Dirac mass on RN giving unit mass to the point 0.

B.1.3 Fundamental solutions

Definition B.1.8. We say that u = u(x, t) is a solution of ∂tu + Lu = 0 in some region Ω if all

derivatives of uwhich occur in ∂tu+Lu = 0 are continuous functions in Ω and ∂tu(x, t)+Lu(x, t) =

0 at each point (x, t) ∈ Ω.

On the other side we define a fundamental solution as

Definition B.1.9. A fundamental solution of ∂tu+ Lu = 0 is a function Γ(x, t; ξ, τ) defined for all

(x, t) ∈ UT , (ξ, τ) ∈ UT , t > τ which satisfies the following conditions:

(i) for fixed (ξ, τ) ∈ UT it satisfies, as a function of (x, t) (x ∈ U , τ < t ≤ T ) the equation

∂tu+ Lu = 0;

B.1 Fundamental solutions for parabolic equations 123

(ii) for every continuous function f(x) ∈ U , if x ∈ U then

limtτ

∫U

Γ(x, t; ξ, τ)f(ξ)dξ = f(x)

We can think of the fundamental solution as a unit source solution: Γ(x, t; ξ, τ) for fixed (ξ, τ)

gives the concentration at the point x at time t, generated by the diffusion of a unit mass initiallyconcentrated at the origin (let us think at remark B.1.7 with initial condition δ0).

From another point of view, if we imagine a unit mass composed of a large number N ofparticles, Γ(x, t; ξ, τ) gives the probability that a single particle is placed between x and ξ at time tor equivalently the percentage of particles inside the interval (x, ξ) at time t. This suggest us that thefundamental solution is the perfect connection between PDE and the transition density function of aprocess associated to the PDE.

Appendix C

Appendix

C.1 The Longstaff Model

The Longstaff model has been proposed by Longstaff as an alternative model to the CIR. It hasbeen shown empirically that this model can fit better data for short rate.

Given the PDE

ut +1

2ρ2x2γuxx + (α+ βx− λρxγ)uu − xu = 0 (C.1)

The solution u(x, t) of this PDE together with the final condition u(x, T ) = 1 gives the price for aZero-Coupon bond with maturity T . The parameter λ is called the market price of risk. It representsan extra increase in the expected rate of return on a bond per additional unit of risk. In general it maydepends upon both x and t.

We will in this section concentrate on the case γ = 12 and α = ρ2

4 which gives us the Longstaffequation.

It is not convenient to proceed with the method developped by Craddock since this equationsolves the first Riccati equation only for certain parameters. This narrow any possible use of thismodel. Anyway another approach is possible. In fact we will show now that this equation can bemapped in the standard heat equation.

We will now look for any possible symmetries of the PDE

ut +1

2ρ2xuxx + (

ρ2

4+ β − λρ

√x)uu − xu = 0 (C.2)

Applying the standar Lie algoritm we can see that the symmetric group for the equation (C.2) is

125

126 C. Appendix

spanned by

v1 = ∂tt

v2 = u∂u

v3 = eµt2

(√x∂x − β−µ

ρ2

(ρλµ +

√x)u∂u

)v4 = e−

µt2

(√x∂x + β+µ

ρ2

(ρλµ −

√x)u∂u

)v5 = e

µt2

(x− βλρ

µ2

√x)∂x + 1

µ∂t + (a0(µ) + a1(µ)√x+ xa2(µ))u∂u)

v6 = eµt2

(x− βλρ

µ2

√x)∂x − 1

µ∂t + (a0(−µ) + a1(−µ)√x+ xa2(−µ))u∂u)

(C.3)

with

µ =√β2 + 2ρ2, a0(ζ) =

β − ζ4ζ

− λ2(ρ2 + (β − ζ)ζ)

ζ3,

a1(ζ) =λ(β − ζ)2

ρζ2, a2(ζ) =

ζ − βρ2

We have already anticipated equation (C.2) can be mapped into the standard heat equation by aninvertible transformation of the form

z = f(x, t), τ = g(t), w = h(x, t)u (C.4)

We have therefore to solve the Cauchy problemwτ = wzz

w(z, 0) = η(ζ)(C.5)

It is well-known that a solution is given by

w(z, τ) =1

2√πτ

∫ ∞−∞

η(ζ) exp

(− (z − ζ)2

4τ

)dζ (C.6)

Obviously the solution obtained in (C.6) is transformable back into a solution of (C.2) by the inverseof (C.4).

In order to construct (C.4) we will use v3 and v5. The solution so found of (C.2) will be thus theZero-Coupon bond price under the Longstaff model.

In particular we have to use

f(x, t) = K1 + e−µt2

(2√x− 2βλρ

µ2

)

g(t) =ρ2(e−µt − e−µT )

2µ

h(x, t) = K2 exp

(Λt+

(β − µ)x

ρ2+

2λ(β − µ)√x

ρµ

)

C.1 The Longstaff Model 127

η(ζ) = exp

(ΛT + (µ− β)

(eµT/2λ(µ+ β)(K1 − ζ)

ρµ2− eµT (K1 − ζ)2

4ρ2− βλ2(β + 2µ)

µ4

))

u(x, t) =eϕ(x,t)√

1 +(β−µ

2 ψ2(T − t))

where K1 and K2 are some constants, Λ = λ2(β2+ρ2)µ2 + µ−β

4 − λ2 and

ϕ(x, t) =1

1 +(β−µ

2 ψ2(T − t)) (2λρψ1(T − t)2

√x+ xψ2(T − t)+

+Λ

2(βψ2(T − t) + µψ3(T − t)) (T − t)− λ2ρ2

µ2

(2βψ1(T − t)2 + ψ2(T − t)

))

and

ψ1(y) =1− e

µy2

µ, ψ2(y) =

1− eµy

µ, ψ3(y) =

1 + eµy

µ

Bibliography

Books

[AS72] Milton Abramowitz and Irene A Stegun. Handbook of Mathematical Functions with

Formulas, Graphs, and Mathematical Tables. National Bureau of Standards Applied

Mathematics Series 55. Tenth Printing. ERIC, 1972.

[BA02] George W. Bluman and Stephen C. Anco. Symmetry and integration methods for differ-

ential equations. Vol. 154. Applied Mathematical Sciences. New York: Springer-Verlag,2002.

[Bau00] Gerd Baumann. Symmetry analysis of differential equations with Mathematicar. With 1CD-ROM (Windows, Macintosh and UNIX). Springer-Verlag–TELOS, New York, 2000.

[BC74] G. W. Bluman and J. D. Cole. Similarity methods for differential equations. AppliedMathematical Sciences, Vol. 13. New York: Springer-Verlag, 1974.

[BCA10] George W. Bluman, Alexei F. Cheviakov, and Stephen C. Anco. Applications of symmetry

methods to partial differential equations. Vol. 168. Applied Mathematical Sciences. NewYork: Springer, 2010. URL: http://dx.doi.org/10.1007/978-0-387-68028-6.

[BK89] George W. Bluman and Sukeyuki Kumei. Symmetries and differential equations. Vol. 81.Applied Mathematical Sciences. New York: Springer-Verlag, 1989.

[BM06] Damiano Brigo and Fabio Mercurio. Interest rate models—theory and practice. Second.Springer Finance. With smile, inflation and credit. Berlin: Springer-Verlag, 2006.

[Bre11] Haim Brezis. Functional analysis, Sobolev spaces and partial differential equations.Universitext. New York: Springer, 2011.

[CZ04] Jakša Cvitanic and Fernando Zapatero. Introduction to the economics and mathematics

of financial markets. Cambridge, MA: MIT Press, 2004.

[EN00] Klaus-Jochen Engel and Rainer Nagel. One-parameter semigroups for linear evolution

equations. Vol. 194. Graduate Texts in Mathematics. New York: Springer-Verlag, 2000.

[ES98] Yu. V. Egorov and M. A. Shubin. Foundations of the classical theory of partial differential

equations. Berlin: Springer-Verlag, 1998.

129

http://dx.doi.org/10.1007/978-0-387-68028-6

http://dx.doi.org/10.1007/978-0-387-68028-6

130 C. Appendix

[Eva10] Lawrence C. Evans. Partial differential equations. Second. Vol. 19. Graduate Studies inMathematics. Providence, RI: American Mathematical Society, 2010.

[Fri64] Avner Friedman. Partial differential equations of parabolic type. Englewood Cliffs, N.J.:Prentice-Hall Inc., 1964.

[Gil08] Robert Gilmore. Lie groups, physics, and geometry. An introduction for physicists,engineers and chemists. Cambridge: Cambridge University Press, 2008.

[Hal03] Brian C. Hall. Lie groups, Lie algebras, and representations. Vol. 222. Graduate Texts inMathematics. An elementary introduction. New York: Springer-Verlag, 2003.

[Hum78] James E. Humphreys. Introduction to Lie algebras and representation theory. Vol. 9.Graduate Texts in Mathematics. Second printing, revised. New York: Springer-Verlag,1978.

[Hyd00] Peter E. Hydon. Symmetry methods for differential equations. Cambridge Texts inApplied Mathematics. A beginner’s guide. Cambridge: Cambridge University Press,2000.

[Ibr+94] N. H. Ibragimov et al. CRC handbook of Lie group analysis of differential equations.

Vol. 1. Symmetries, exact solutions and conservation laws. Boca Raton, FL: CRC Press,1994.

[Ibr+95] N. H. Ibragimov et al. CRC handbook of Lie group analysis of differential equations. Vol.

2. Applications in engineering and physical sciences, Edited by Ibragimov. Boca Raton,FL: CRC Press, 1995.

[Ibr+96] N. H. and Ibragimov et al. CRC handbook of Lie group analysis of differential equations.

Vol. 3. New trends in theoretical developments and computational methods, With 1IBM-PC floppy disk (3.5 inch; HD). Boca Raton, FL: CRC Press, 1996.

[Ibr09] N. H. Ibragimov. Transformation groups and Lie algebras. 2009. URL: http://gammett.ugatu.su/images/docs/publ_ibr/tr_groups_nhi.pdf.

[Jos08] M. S. Joshi. The concepts and practice of mathematical finance. Second. Mathematics,Finance and Risk. Cambridge: Cambridge University Press, 2008.

[LL08] Damien Lamberton and Bernard Lapeyre. Introduction to stochastic calculus applied

to finance. Second. Chapman & Hall/CRC Financial Mathematics Series. Chapman &Hall/CRC, Boca Raton, FL, 2008.

[LLL10] Cheng-Few Lee, Alice C Lee, and John Lee. Handbook of quantitative finance and risk

management. Springer, 2010.

[Olv93] Peter J. Olver. Applications of Lie groups to differential equations. Second. Vol. 107.Graduate Texts in Mathematics. New York: Springer-Verlag, 1993. URL: http://dx.doi.org/10.1007/978-1-4612-4350-2.

[Ovs82] L. V. Ovsiannikov. Group analysis of differential equations. New York: Academic PressInc. [Harcourt Brace Jovanovich Publishers], 1982.

http://gammett.ugatu.su/images/docs/publ_ibr/tr_groups_nhi.pdf

http://gammett.ugatu.su/images/docs/publ_ibr/tr_groups_nhi.pdf

http://dx.doi.org/10.1007/978-1-4612-4350-2

http://dx.doi.org/10.1007/978-1-4612-4350-2


[Par0 ] di Elea Parmenide. Φερι Φυσεωζ, Peri Physeos. 500 a.C.

[Pas11a] Andrea Pascucci. PDE and martingale methods in option pricing. Vol. 2. Bocconi &Springer Series. Milan: Springer, 2011. URL: http://dx.doi.org/10.1007/978-88-470-1781-8.

[Pas11b] Andrea Pascucci. PDE and martingale methods in option pricing. Vol. 2. Bocconi &Springer Series. Milan: Springer, 2011. URL: http://dx.doi.org/10.1007/978-88-470-1781-8.

[Sal08] Sandro Salsa. Partial differential equations in action. Universitext. From modelling totheory. Springer-Verlag Italia, Milan, 2008.

[Shr04] Steven E. Shreve. Stochastic calculus for finance. II. Springer Finance. Continuous-timemodels. New York: Springer-Verlag, 2004.

[Ste89] Hans Stephani. Differential equations. Their solution using symmetries. Cambridge:Cambridge University Press, 1989.

[Var84] V. S. Varadarajan. Lie groups, Lie algebras, and their representations. Vol. 102. GraduateTexts in Mathematics. Reprint of the 1974 edition. New York: Springer-Verlag, 1984.

Papers

[BL10] DR Brecher and AE Lindsay. “Results on the CEV process, past and present”. In:preprint (2010).

[CD01] M. J. Craddock and A. H. Dooley. “Symmetry group methods for heat kernels”. In: J.

Math. Phys. 42.1 (2001), pp. 390–418. URL: http://dx.doi.org/10.1063/1.1316763.

[CD10] M. J. Craddock and A. H. Dooley. “On the equivalence of Lie symmetries and grouprepresentations”. In: J. Differential Equations 249.3 (2010), pp. 621–653. URL: http://dx.doi.org/10.1016/j.jde.2010.02.003.

[CDPep] F.G. Cordoni and L. Di Persio. “Transition density for the CIR process by Lie symme-tries and application to ZCB pricing”. In: International Journal of Pure and Applied

Mathematics (preprint).

[CHL10] HW Chuang, YL Hsu, and CF Lee. “Application of the Characteristic Function inFinancial Research”. In: Handbook of Quantitative Finance and Risk Management.Springer, 2010.

[CIR85] John C. Cox, Jonathan E. Ingersoll Jr., and Stephen A. Ross. “A theory of the termstructure of interest rates”. In: Econometrica 53.2 (1985). URL: http://dx.doi.org/10.2307/1911242.

[CL07] Mark Craddock and Kelly A. Lennox. “Lie group symmetries as integral transforms offundamental solutions”. In: J. Differential Equations 232.2 (2007), pp. 652–674. ISSN:0022-0396. URL: http://dx.doi.org/10.1016/j.jde.2006.07.011.

http://dx.doi.org/10.1007/978-88-470-1781-8

http://dx.doi.org/10.1007/978-88-470-1781-8

http://dx.doi.org/10.1007/978-88-470-1781-8

http://dx.doi.org/10.1007/978-88-470-1781-8

http://dx.doi.org/10.1063/1.1316763

http://dx.doi.org/10.1063/1.1316763

http://dx.doi.org/10.1016/j.jde.2010.02.003


http://dx.doi.org/10.2307/1911242

http://dx.doi.org/10.2307/1911242


132 C. Appendix

[CL09] Mark Craddock and Kelly A. Lennox. “The calculation of expectations for classes ofdiffusion processes by Lie symmetry methods”. In: Ann. Appl. Probab. 19.1 (2009),pp. 127–157. URL: http://dx.doi.org/10.1214/08-AAP534.

[CL10] Ren-Raw Chen and Cheng-Few Lee. “A Constant Elasticity of Variance (CEV) Familyof Stock Price Distributions in Option Pricing, Review, and Integration”. In: Handbook

of Quantitative Finance and Risk Management. Springer, 2010.

[CL12] Mark Craddock and Kelly A. Lennox. “Lie symmetry methods for multi-dimensionalparabolic PDEs and diffusions”. In: J. Differential Equations 252.1 (2012), pp. 56–90.URL: http://dx.doi.org/10.1016/j.jde.2011.09.024.

[COG10] Nicolette C. Caister, John G. O’Hara, and Keshlan S. Govinder. “Solving the Asianoption PDE using Lie symmetry methods”. In: Int. J. Theor. Appl. Finance 13.8 (2010),pp. 1265–1277. URL: http://dx.doi.org/10.1142/S0219024910006194.

[CP03] Mark Craddock and Eckhard Platen. Symmetry group methods for fundamental solu-

tions and characteristic functions. School of Finance and Economics, University ofTechnology, Sydney, 2003.

[CP04] Mark Craddock and Eckhard Platen. “Symmetry group methods for fundamental so-lutions”. In: J. Differential Equations 207.2 (2004), pp. 285–302. URL: http://dx.doi.org/10.1016/j.jde.2004.07.026.

[CP09] Mark Craddock and Eckhard Platen. “On explicit probability laws for classes of scalardiffusions”. In: Quantitative Finance Research Centre Research Paper 246 (2009).

[Cra09] Mark Craddock. “Fundamental solutions, transition densities and the integration of Liesymmetries”. In: J. Differential Equations 246.6 (2009), pp. 2538–2560. URL: http://dx.doi.org/10.1016/j.jde.2008.10.017.

[Cra94] Mark Craddock. “Symmetry groups of partial differential equations, separation of vari-ables, and direct integral theory”. In: J. Funct. Anal. 125.2 (1994), pp. 452–479. URL:http://dx.doi.org/10.1006/jfan.1994.1133.

[Cra95] Mark Craddock. “The symmetry groups of linear partial differential equations andrepresentation theory. I”. In: J. Differential Equations 116.1 (1995), pp. 202–247. URL:http://dx.doi.org/10.1006/jdeq.1995.1034.

[Fel51] William Feller. “Two singular diffusion problems”. In: Ann. of Math. (2) 54 (1951),pp. 173–182.

[GI98] R. K. Gazizov and N. H. Ibragimov. “Lie symmetry analysis of differential equationsin finance”. In: Nonlinear Dynam. 17.4 (1998), pp. 387–407. ISSN: 0924-090X. DOI:10.1023/A:1008304132308. URL: http://dx.doi.org/10.1023/A:1008304132308.

[Goa00] J. Goard. “New solutions to the bond-pricing equation via Lie’s classical method”. In:Math. Comput. Modelling 32.3-4 (2000), pp. 299–313. URL: http://dx.doi.org/10.1016/S0895-7177(00)00136-9.

http://dx.doi.org/10.1214/08-AAP534


http://dx.doi.org/10.1142/S0219024910006194





http://dx.doi.org/10.1006/jfan.1994.1133

http://dx.doi.org/10.1006/jdeq.1995.1034

http://dx.doi.org/10.1023/A:1008304132308

http://dx.doi.org/10.1023/A:1008304132308

http://dx.doi.org/10.1023/A:1008304132308

http://dx.doi.org/10.1016/S0895-7177(00)00136-9

http://dx.doi.org/10.1016/S0895-7177(00)00136-9


[Goa06] Joanna Goard. “Fundamental solutions to Kolmogorov equations via reduction to canon-ical form”. In: J. Appl. Math. Decis. Sci. (2006), Art. ID 19181, 24. URL: http://dx.doi.org/10.1155/JAMDS/2006/19181.

[Gov01] K. S. Govinder. “Lie subalgebras, reduction of order, and group-invariant solutions”. In:J. Math. Anal. Appl. 258.2 (2001), pp. 720–732. URL: http://dx.doi.org/10.1006/jmaa.2001.7513.

[Hes93] Steven L Heston. “A closed-form solution for options with stochastic volatility withapplications to bond and currency options”. In: Review of financial studies 6.2 (1993),pp. 327–343.

[HLL08] Y. L. Hsu, T. I. Lin, and C. F. Lee. “Constant elasticity of variance (CEV) option pricingmodel: integration and detailed derivation”. In: Math. Comput. Simulation 79.1 (2008),pp. 60–71. URL: http://dx.doi.org/10.1016/j.matcom.2007.09.012.

[Ibr94] N. H. Ibragimov. “Lie group analysis of differential equations. A mosaic of results andopen problems”. In: The Sophus Lie Memorial Conference (Oslo, 1992). Oslo: Scand.Univ. Press, 1994, pp. 53–75.

[Lie80] Sophus Lie. “Theorie der transformationsgruppen I”. In: Mathematische Annalen 16.4(1880), pp. 441–528.

[LYH00] Chi Fai LO, PH Yuen, and CH Hui. “Constant elasticity of variance option pricing modelwith time-dependent parameters”. In: International Journal of Theoretical and Applied

Finance 3.04 (2000), pp. 661–674.

[SLO08a] W. Sinkala, P. G. L. Leach, and J. G. O’Hara. “Invariance properties of a general bond-pricing equation”. In: J. Differential Equations 244.11 (2008), pp. 2820–2835. URL:http://dx.doi.org/10.1016/j.jde.2008.02.044.

[SLO08b] W. Sinkala, P. G. L. Leach, and J. G. O’Hara. “Zero-coupon bond prices in the Vasicekand CIR models: their computation as group-invariant solutions”. In: Math. Methods Appl.

Sci. 31.6 (2008), pp. 665–678. URL: http://dx.doi.org/10.1002/mma.935.

http://dx.doi.org/10.1155/JAMDS/2006/19181

http://dx.doi.org/10.1155/JAMDS/2006/19181

http://dx.doi.org/10.1006/jmaa.2001.7513

http://dx.doi.org/10.1006/jmaa.2001.7513

http://dx.doi.org/10.1016/j.matcom.2007.09.012


http://dx.doi.org/10.1002/mma.935

Index

Φερι Φυσεωζ, Peri Physeos, 131

A closed-form solution for options with stochastic

volatility with applications to bond and

currency options, 108, 133

A Constant Elasticity of Variance (CEV) Fam-

ily of Stock Price Distributions in Op-

tion Pricing, Review, and Integration,vi, 100, 132

A theory of the term structure of interest rates,131

Abramowitz, Milton, 129

Anco, Stephen C., v, 58, 62, 129

Application of the Characteristic Function in Fi-

nancial Research, vi, 100, 131

Applications of symmetry methods to partial dif-

ferential equations, v, 58, 62, 129

Applications of Lie groups to differential equa-

tions, v, 24, 32, 130

Baumann, Gerd, 129

Bluman, G. W., v, 129

Bluman, George W., v, 52, 58, 62, 129

Brecher, DR, 131

Brezis, Haim, 129

Brigo, Damiano, vi, 91, 92, 107, 129

Caister, Nicolette C., 132

Chen, Ren-Raw, vi, 100, 132

Cheviakov, Alexei F., v, 58, 62, 129

Chuang, HW, vi, 100, 131

Cole, J. D., v, 129

Constant elasticity of variance (CEV) option pric-

ing model: integration and detailed

derivation, vi, 100, 133

Constant elasticity of variance option pricing

model with time-dependent parameters,133

Cordoni, F.G., 131Cox, John C., 131Craddock, M. J., 131Craddock, Mark, vi, 78, 82, 83, 91, 131, 132Cvitanic, Jakša, vi, 99, 129CRC handbook of Lie group analysis of differen-

tial equations. Vol. 1, v, 130CRC handbook of Lie group analysis of differen-

tial equations. Vol. 2, v, 130CRC handbook of Lie group analysis of differen-

tial equations. Vol. 3, v, 130

Differential equations, 131Di Persio, L., 131Dooley, A. H., 131

Egorov, Yu. V., 129Engel, Klaus-Jochen, 129Evans, Lawrence C., 74, 119, 130

Feller, William, 99, 132Foundations of the classical theory of partial dif-

ferential equations, 129Friedman, Avner, vi, 74, 92, 119, 130Functional analysis, Sobolev spaces and partial

differential equations, 129Fundamental solutions to Kolmogorov equations

via reduction to canonical form, 133Fundamental solutions, transition densities and

the integration of Lie symmetries, vi,82, 91, 132

Gazizov, R. K., 132

134

INDEX 135

Gilmore, Robert, 130Goard, J., 132Goard, Joanna, 133Govinder, K. S., 133Govinder, Keshlan S., 132Group analysis of differential equations, iv, v,

130

Hall, Brian C., v, 130Handbook of Mathematical Functions with For-

mulas, Graphs, and Mathematical Ta-

bles. National Bureau of Standards

Applied Mathematics Series 55. Tenth

Printing., 129Handbook of quantitative finance and risk man-

agement, 130Heston, Steven L, 108, 133Hsu, Y. L., vi, 100, 133Hsu, YL, vi, 100, 131Hui, CH, 133Humphreys, James E., 130Hydon, Peter E., 130

Ibragimov, N. H., v, 130, 132, 133Ibragimov, N. H. and, v, 130Ingersoll Jr., Jonathan E., 131Interest rate models—theory and practice, vi, 91,

92, 107, 129Introduction to stochastic calculus applied to fi-

nance, vi, 130Introduction to the economics and mathematics

of financial markets, vi, 99, 129Introduction to Lie algebras and representation

theory, 130Invariance properties of a general bond-pricing

equation, vi, 133

Joshi, M. S., 107, 130

Kumei, Sukeyuki, v, 52, 58, 62, 129

Lamberton, Damien, vi, 130Lapeyre, Bernard, vi, 130

Leach, P. G. L., vi, 133Lee, Alice C, 130Lee, C. F., vi, 100, 133Lee, CF, vi, 100, 131Lee, Cheng-Few, vi, 100, 130, 132Lee, John, 130Lennox, Kelly A., vi, 82, 83, 91, 131, 132Lie group analysis of differential equations. A

mosaic of results and open problems, v,133

Lie group symmetries as integral transforms of

fundamental solutions, vi, 82, 131Lie groups, physics, and geometry, 130Lie groups, Lie algebras, and representations, v,

130Lie groups, Lie algebras, and their representa-

tions, v, 131Lie subalgebras, reduction of order, and group-

invariant solutions, 133Lie symmetry analysis of differential equations in

finance, 132Lie symmetry methods for multi-dimensional parabolic

PDEs and diffusions, vi, 132Lie, Sophus, 133Lin, T. I., vi, 100, 133Lindsay, AE, 131LO, Chi Fai, 133

Mercurio, Fabio, vi, 91, 92, 107, 129

Nagel, Rainer, 129New solutions to the bond-pricing equation via

Lie’s classical method, 132

O’Hara, J. G., vi, 133O’Hara, John G., 132Olver, Peter J., v, 24, 32, 130On explicit probability laws for classes of scalar

diffusions, vi, 132On the equivalence of Lie symmetries and group

representations, 131One-parameter semigroups for linear evolution

equations, 129

136 INDEX

Ovsiannikov, L. V., iv, v, 130

Parmenide, di Elea, 131Partial differential equations, 74, 119, 130Partial differential equations in action, 131Partial differential equations of parabolic type,

vi, 74, 92, 119, 130Pascucci, Andrea, vi, 131Platen, Eckhard, vi, 78, 132PDE and martingale methods in option pricing,

vi, 131

Results on the CEV process, past and present, 131Ross, Stephen A., 131

Salsa, Sandro, 131Shreve, Steven E., vi, 99, 131Shubin, M. A., 129Similarity methods for differential equations, v,

129Sinkala, W., vi, 133Solving the Asian option PDE using Lie symmetry

methods, 132Stegun, Irene A, 129Stephani, Hans, 131Stochastic calculus for finance. II, vi, 99, 131Symmetries and differential equations, v, 52, 58,

62, 129Symmetry and integration methods for differential

equations, v, 129Symmetry group methods for fundamental solu-

tions, vi, 78, 132Symmetry group methods for fundamental solu-

tions and characteristic functions, vi,132

Symmetry group methods for heat kernels, 131Symmetry groups of partial differential equations,

separation of variables, and direct inte-

gral theory, 132Symmetry methods for differential equations, 130

The calculation of expectations for classes of dif-

fusion processes by Lie symmetry meth-

ods, vi, 82, 83, 91, 132The concepts and practice of mathematical fi-

nance, 107, 130The symmetry groups of linear partial differential

equations and representation theory. I,132

Theorie der transformationsgruppen I, 133Transformation groups and Lie algebras, v, 130Transition density for the CIR process by Lie sym-

metries and application to ZCB pricing,131

Two singular diffusion problems, 99, 132

Varadarajan, V. S., v, 131

Yuen, PH, 133

Zapatero, Fernando, vi, 99, 129Zero-coupon bond prices in the Vasicek and CIR

models: their computation as group-

invariant solutions, vi, 133

lie symmetry method for partial differential equations ... · differential equations with fewer...

Documents