stochastic diﬀerential equations (sdes) · stochastic diﬀerential equations constitute a very...

Stochastic Di!erential Equations (SDEs)

Francesco Russo

Universite Paris 13, Institut Galilee, Mathematiques

99, av. JB Clement, F-99430 Villetaneuse, France

October 4th 2005

1 Introduction

This paper will appear in Encyclopedia of Mathematical Physics, see

http://www1.elsevier.com/homepage/about/mrwd/emp/menu.htm.

and it constitutes a survey paper on the title subject.

Stochastic di!erential equations (SDEs) appear today as a modeling

tool in several sciences as telecommunications, economics, finance, bi-

ology and quantum field theory.

A stochastic di!erential equation is essentially a classical di!erential

equation which is perturbed by a random noise. When nothing else is

specified, SDE means in fact ordinary stochastic di!erential equation;

in that case it corresponds to the perturbation of an ordinary di!er-

ential equation. Stochastic partial di!erential equations (SPDEs) are

obtained as random perturbation of partial di!erential equations.

One of the most important di!erence between deterministic and stochas-

tic ordinary di!erentials equation is described by the so called Peano

type phenomenon. A classical di!erential equation with continuous and

linear growth coe"cients admits global existence but not uniqueness as

1

Stochastic Di!erential Equations. 2

classical calculus text books illustrate studying equations of the type

dX

dt(t) =

!X(t), X(0) = 0.

However if one perturbes the right member of the equality with an

additive Gaussian white noise (!t) (even with very small intensity),

then the problem becomes well-stated. A similar phenomenon happens

with linear PDEs of evolution type perturbed with a space-time white

noise.

Stochastic di!erential equations constitute a very huge subject and an

incredible amount of relevant contributions were brought. We try to

orientate the reader about the main axes trying to indicate references

to the di!erent subfields. We will prefer to refer to monographies when

available, instead of articles.

2 Motivation and preliminaries

In the whole paper T will be a strictly positive real number. Let us

consider continuous functions b : R+ ! Rd " Rd, a : R+ ! Rd!m " Rd

and x0 # Rd. We consider a di!erential problem of the following type."

dXtdt = b(t, Xt)

X0 = x0.(2.1)

Let (#,F , P) be a complete probability space. Suppose that previous

equation is perturbed by a random noise (!t)t"0. Because of modeling

reasons it could be reasonnable to suppose (!t)t"0 verifying the following

properties.

1) It is a family of independent random variables.

2) (!t)t"0 is stationary i.e. for any positive integer n, positive reals

h, t0, t1, . . . , tn the law of (!t0+h, . . . , !tn+h) does not depend on h.


More precisely we perturb previous equation (2.1) as follows:

"dXtdt = b(t, Xt) + a(t, Xt)!t

X0 = x0.(2.2)

We suppose for a moment that d = m = 1. In reality no reasonnable

real valued process (!t)t"0 fulfilling previous assumptions exists. In

particular, if process (!t) exists (resp. exists and each !t is a square

integrable random variable) then the process cannot have continuous

paths (resp. it cannot be measurable with respect to #!R+). However,

suppose that such a process exists; we set Bt =# t

0 !sds. In that case,

previous properties 1) and 2) can be translated into the following on

(Bt).

(i) It has independent increments, that means that for any t0, . . . , tn, h $0, Bt1+h % Bt0+h, . . . , Btn+h % Btn!1+h are independent random

variables.

(ii) It has stationary increments, that means that for any t0, . . . , tn, h $0, the law of (Bt1+h%Bt0+h, . . . , Btn+h%Btn!1+h) does not depend

on h.

On the other hand it is natural to require that

a) B0 = 0 a.s.,

b) it is a continuous process, i.e. it has continuous paths a.s.

Equation (2.2) should be rewritten in some integral form

Xt = X0 +

$ t

0

b(s, Xs)ds +

$ t

0

a(s, Xs)dBs. (2.3)

Clearly the paths of process (Bt) cannot be di!erentiable so one has to

give meaning to integral# t

0 a(s, Xs)dBs . This will be intended in the

Ito sense, see considerations below.


An important result of probability theory says that a stochastic process

(Bt) fulfilling properties (i), (ii) and a) b) is essentially a Brownian

motion. More precisely, there are real constants b, " such that Bt =

bt + "Wt where (Wt) is a classical Brownian motion defined below.

Definition 2.1 (i) A (continuous) stochastic process (Wt) is called

classical Brownian motion if W0 = 0 a.s., it has independent

increments and the law of Wt % Ws is a Gaussian N(0, t % s)

random variable.

(ii) A m-dimensional Brownian motion is a vector (W 1, . . . , W m)

of independent classical Brownian motions.

Let (Ft)t"0 be a filtration fulfilling the usual conditions, see Section

1.1 of [9]. There one can find basic concepts of the theory of stochas-

tic processes as the concept of adapted, progressively measurable

process. An adapted process is also said to be non-anticipating towards

the filtration (Ft) which represents the state of the information at each

time t. A process (Xt) is said to be adapted if for any t, Xt is Ft-

measurable. The notion of progressively measurable process is a slight

refinement of the notion of adapted process.

Definition 2.2 (i) A (continuous) (Ft) adapted process (Wt) is called

(classical) (Ft)-Brownian motion if W0 = 0, if for any s < t

Wt % Ws is an N(0, t % s) distributed random variable which is

independent of Fs.

(ii) An (Ft)-m-dimensional Brownian motion is a vector (W 1, . . . , W m)

of (Ft)-classical independent Brownian motions.

From now on we will consider a probability space (#,F , P) equipped

with a filtration (Ft)t"0 fulfilling the usual conditions. From now on all

the considered filtrations will have that property.


Let W = (Wt)t"0 be an (Ft)t"0- m-dimensional classical Brownian mo-

tion. In chapter 3 of [9] and chapter 4 of [17] one introduces the notion

of stochastic Ito integral announced before. Let Y = (Y 1, . . . , Y m) be a

progressively measurable m- dimensional process such that# T

0 &Ys&2ds <

', then the Ito integral# T

0 YsdWs is well defined. In particular the in-

definite integral.# ·0 YsdWs is an (Ft)-progressively measurable continu-

ous process. If Y is a Rd!m matrix-valued process, the integral# t0 YsdWs

is componentwise defined and it will be a vector in Rd. The analogous

of di!erential calculus in the framework of stochastic processes, is Ito

calculus, see again [9] ch. 3 and ch. IV of [17]. Important tools

are the concept of quadratic variation [X] of a stochastic process

when it exists. For instance the quadratic variation [W ]t of a classical

Brownian motion equals t. If Mt =# t

0 YsdWs then [M ]t =# t

0 &Ys&2ds.

One celebrated theorem of P. Levy states the following: if (Mt) defines

a continuous (Ft)-local martingale such that [Mt] ( t, then M is an

(Ft)-classical Brownian motion. That theorem is called the Levy car-

acterization theorem of Brownian motion. Ito formula constitutes the

natural generalization of fundamental theorem of di!erential calculus

to the stochastic calculus. Another significant tool is Girsanov theo-

rem; it states essentially the following: suppose that the following so

called Novikov condition is verified:

E%

exp

%1

2

$ T

0

&Yt&2

&dt

&< '.

Then the process Wt = Wt+# t

0 Ysds, t # [0, T ] is again a m-dimensional

(Ft)- classical Brownian motion under a new probability measure Q on

(#,FT ) defined by dQ = dP exp'# t

0 YsdWs % 12&Ys&2ds

(.

Let ! be an F0-measurable random variable, for instance ! ( x # Rd.

We are interested in the stochastic di!erential equation"

dXt = a(t, Xt)dWt + b(t, Xt)dt

X0 = !.(2.4)


Definition 2.3 A progressively measurable process (Xt)t#[0,T ] is said to

be solution of (2.4) if a.s.

Xt = Z +

$ t

0

a(t, Xt)dWt +

$ t

0

b(t, Xt)dt, )t # [0, T ]. (2.5)

provided that the right member makes sense. In particular, such a so-

lution is continuous. The function a (resp. b) is called the di!usion

(drift) coe"cient of the SDE. a and b may sometimes be allowed to

be random; however this dependence has to be progressively measur-

able. Clearly, we can define the notion of solution (Xt)t"0 on the whole

positive real axis.

We remark that those equations are called Ito stochastic di!eren-

tial equations. A solution of previous equation is named di!usion

process.

3 The Lipschitz case

The most natural framework for studying existence and uniqueness

for stochastic di!erential equations appears when the coe"cients are

Lipschitz.

A function # : [0, T ]!Rm %" Rd is said to have polynomial growth

(with respect to x uniformly in t), if for some n there is a constant

C > 0 with

supt#[0,T ]

&#(t, x)& * C(1 + &x&n) (3.6)

The same fonction is said to have linear growth if (3.6) holds with n =

1. A function # : R+ !Rm " Rd is said to be locally Lipschitz (with

respect to x uniformly in t), if for every t # [0, T ], K > 0, #|[0,T ]![$K,K]

is Lipschitz (with respect to x uniformly with respect to t).

Let a : R+ ! Rd!m %" Rd, b : R+ ! Rd %" Rd, be Borel functions,

! an Rd-valued random variable F0-measurable and (Wt)t"0 be a m-

dimensional (Ft)-Brownian motion.


Classical fixed point theorems allow to establish the following classical

result.

Theorem 3.1 We suppose a and b locally Lipschitz with linear growth.

Let ! be a square integrable r.v. that is F0-measurable. Then (2.4) has

a unique solution X. Moreover,

E%

supt%T

|Xt|2&

< '.

Remark 3.2 (i) Equation (2.4) can be settled similarly by putting

initial condition x at some time s. In that case the problem is

again well stated. If ! ( x is a deterministic point of Rd than we

will often denote by Xs,x the solution of that problem.

(ii) If the coe"cients are only locally Lipschitz, the equation may be

solved until a stopping time. If d = 1 it is possible to state neces-

sary and su"cient conditions for non-explosion (Feller test).

(iii) The theorem above admits several generalizations. For instance

the Brownian motion can be replaced by a general semimartin-

gales, (possibly with jumps as Levy processes).

An important role of di!usion processes is the fact that they provide

probabilistic representation to partial di!erential equations of parabolic

(and even elliptic) type. We will only mention here the parabolic frame-

work.

We denote A(t, x) = a(t, x)a(t, x)& where + means transposition for

matrices. (t, x) " A(t, x) = (Aij(t, x)) is a d!d matrix valued function.

Let us consider also continuous functions k : [0, T ] ! Rd " Rd, g :

[0, T ] ! Rd " Rd with polynomial growth or non-negative.

Given a solution of (3.1), we can associate its generator (Lt, t # [0, T ])

setting

Ltf(x) =1

2

d)

i,j=1

Aij(t, x)$2ijf(x) + b(t, x) ·,f(x).


Feynmann-Kac theorem allows to represent probabilistically as follows

Theorem 3.3 Suppose there is a function v : [0, T ] ! Rd " Rd con-

tinuous with polynomial growth of class C1,2([0, T [!Rd) satisfying the

following Cauchy problem"

($tv + Lt)v % kv = g

v(T, x) = f(x).(3.7)

Then

v(s, x) = E

%f(XT ) exp(%

$ T

s

k(%, X!)d%) %$ T

s

g(t, Xt) exp{%$ t

s

k(%, X!)d%)}dt

&

for (s, x) # [0, T ] ! Rd where X = Xs,x. In particular, such a solution

is unique.

Remark 3.4 (i) In order to obtain “classical solutions” of previous

Cauchy problem one needs some conditions. It is the case for

instance when the following ellipticity condition holds on A:

-c > 0, )(t, x) # [0, T ] ! Rn

(3.8)

)(!1, . . . , !n) # Rn :d)

i,j

Aij(t, x)!i!j $ cd)

i=1

|!i|2.

In the degenerate case it is possible to deal with viscosity solutions,

in the sense of P.L. Lions. This theorem establishes an impor-

tant link between deterministic PDEs and stochastic di!erential

equations.

(ii) A natural generalization of Feynmann-Kac theorem comes from

the system of forward-backward stochastic di!erential equations

in the sense of Pardoux and Peng.

(iii) Other types of probabilistic representation do appear in stochastic

control theory through the so-called verification theorems, see for


instance [6, 20]. In that case the (non-linear) Hamilton-Jacobi-

Bellmann deterministic equation is represented by a controlled

stochastic di!erential equation.

(iv) Another bridge between non-linear PDEs and di!usions can be

provided in the framework of interacting particle systems with

chaos propagation, see [7] for a survey on those problems. Among

the most significant non-linear PDEs investigated probabilistically,

we quote the case of porous media equations. For instance, for a

positive integer m, a solution to

$tu =1

2$2

xx(u2m+1) (3.9)

can be represented by a (non-linear) di!usion of the type, see [1],"

dXt = um(s, Xs)dWt

u(t, ·) = law density of Xt.(3.10)

4 Di!erent notions of solutions

Let a and b as at the beginning of section 3. Let (#,F , P) be a probabil-

ity space, a filtration (Ft)t"0 fulfilling the usual conditions, an (Ft)t"0-

classical Brownian motion (Wt)t"0. Let ! be an F0-measurable random

variable. At section 2 we defined the notion of solution of the following

equation: "dXt = b(t, Xt)dt + a(t, Xt)dWt

X0 = !.(4.11)

Previous equation will be denoted by E(a, b) (without initial condition).

However, as we will see, the general concept of solution of an SDE

is more sophisticated and subtle than in the deterministic case. We

distinguish several variants of existence and uniqueness.

Definition 4.1 (Strong existence)

We will say that equation E(a, b) admits strong existence if the fol-

lowing holds. Given any probability space (#,F , P), a filtration (Ft)t"0,


an (Ft)t"0- Brownian motion (Wt)t"0, a F0-measurable and square in-

tegrable random variable !, there is a process (Xt)t"0 solution to E(a, b)

with X0 = ! a.s.

Definition 4.2 (Pathwise uniqueness) We will say that equation

E(a, b) admits pathwise uniqueness if the following property is fulfilled.

Let (#,F , P) be a probability space, a filtration (Ft)t"0, an (Ft)t"0

Brownian motion (Wt)t"0. If two processes X, X are two solutions such

that X0 = X0 a.s., then X and X coincide.

Definition 4.3 (Existence in law or weak existence) Let & be a

probability law on Rd. We will say that E(a, b; &) admits weak existence

if there is a probability space (#,F , P), a filtration (Ft)t"0, an (Ft)t"0-

Brownian motion (Wt)t"0 and a process (Xt)t"0 solution of E(a, b) with

& being the law of X0.

We say that E(a, b) admits weak existence if E(a, b; &) admits weak

existence for every &.

Definition 4.4 (Uniqueness in law) Let & be a probability law on Rd.

We say that E(a, b; &) has a unique solution in law if the following

holds. We consider an arbitrary probability space (#,F , P) and a filtra-

tion (Ft)t"0 on it; we consider also another probability space (#, F , P)

equipped with another filtration (Ft)t"0; we consider an (Ft)t"0-Brownian

motion (Wt)t"0, and an (Ft)t"0- Brownian motion (Wt)t"0; we suppose

having a process (Xt)t"0 (resp. a process (Xt)t"0) solution of E(a, b) on

the first (resp. on the second) probability space such that both the law

of X0 and X0 are identical to &. Then X and X must have the same

law as r.v. with values in E = C(R+) (or C[0, T ]).

We say that E(a, b) has a unique solution in law if E(a, b; &) has a

unique solution in law for every &.

There are important theorems which establish bridges among the pre-

ceding notions. One of the most celebrated is the following.


Proposition 4.5 (Yamada-Watanabe) Consider the equation E(a, b).

i) Pathwise uniqueness implies uniqueness in law.

ii) Weak existence and pathwise uniqueness imply strong existence.

A version can be stated for E(a, b; &) where & is a fixed probability law.

Remark 4.6 i) If a and b are locally Lipschitz with linear growth,

Theorem 3.1 implies that E(a, b) admits strong existence and path-

wise uniqueness.

ii) If a and b are only locally Lipschitz, then pathwise uniqueness is

fulfilled.

4.1 Existence and uniqueness in law

A way to create weak solutions of E(1, b) when (t, x) " b(t, x) is Borel

with linear growth is Girsanov theorem. Suppose d = 1 for simplicity.

Let us consider an (Ft)-classical Brownian motion (Xt). We set

Wt = Xt %$ t

0

b(s, Xs)ds.

Under some suitable probability Q, (Wt) is an (Ft)- classical Brownian

motion. Therefore (Xt) provides a solution to E(1, b; '0).

We continue with an example where E(a, b) does not admit pathwise

uniqueness, even though it admits uniqueness in law.

Example 4.7 We consider the stochastic equation

Xt =

$ t

0

sign(Xs)dWs. (4.12)

with

sign(x) =

"1 if x $ 0,

%1 if x < 0,

It corresponds to E(a, b; '0) with b = 0 and a(x) = sign(x).


If (Wt)t"0 is an (Ft)- classical Brownian motion then (Xt)t"0 is (Ft)t"0-

continuous local martingale vanishing at zero such that [X]t ( t. Ac-

cording to Levy caracterization theorem stated in section 2, X is an

(Ft)t"0-classical Brownian motion. This shows in particular that E(a, b; '0)

admits uniqueness in law. In the sequel we will show that E(a, b; '0)

also admits weak existence.

Let now (#,F , P) be a probability space, an (Ft)t"0- classical Brown-

ian motion with respect to a filtration and (Xt)t"0 such that (4.12) is

verified. Then Xt = %Xt can be shown to be also a solution. Therefore

E(a, b; '0) does not admit pathwise uniqueness.

We continue stating a result true in the multi-dimensional case.

Proposition 4.8 (Stroock-Varadhan). Let & be a probability on Rd

such that $

R&x&2m&(dx) < +', (4.13)

for a certain m > 1. We suppose that a, b are continuous with linear

growth. Then E(a, b; &) admits weak existence.

From now on, a function # : [0, T ] ! Rm " Rd will be said Holder

continuous if it is Holder continuous in the space variable x # Rm

uniformly with respect to the time variable t # [0, T ].

From the same authors, an easy consequence of Th. 7.2.1 in [18] is the

following result.

Proposition 4.9 We suppose a, b both Holder continuous, bounded

such that conditon (3.8) is fulfilled. Then SDE E(a, b; &) admits weak

uniqueness.

Remark 4.10 (i) The Holder condition and (3.8) in Proposition 4.9

may be relaxed and replaced with the solvability of a Cauchy prob-

lem of a parabolic PDE with suitable terminal value.


(ii) In the case d = 1, if a, b are bounded and just Borel with (3.8)

for x on each compact, then E(a, b; &) admits weak existence and

uniqueness in law. See [18] exercises 7.3.2 and 7.3.3.

(iii) If d = 2, the same holds as at previous point provided that more-

over a does not depend on time.

We proceed with some more specifically unidimensional material stating

some results from K.J. Engelbert and W. Schmidt. These two authors

furnished necessary and su"cient conditions for weak existence and

uniqueness in law of SDEs.

For a Borel function " : R " R, we first define

Z(") = {x # R|"(x) = 0};

then we define the set I(") as the set of real numbers x such that$ x+"

x$"

dy

"2(y)= ', )( > 0.

Proposition 4.11 (Engelbert-Schmidt criterion).

Suppose that a : R " R, i.e. does not depend on time and we consider

the equation without drift E(a, 0).

i) E(a, 0) admits weak existence (without explosion) if and only if

I(a) . Z(a) (4.14)

ii) E(a, 0) admits weak existence and uniqueness in law if and only if

I(a) = Z(a) (4.15)

Remark 4.12 i) If a is continuous then (4.14) is always verified.

Indeed, if a(x) /= 0, there is ( > 0 such that

|a(y)| > 0, )y # [x % (, x + (].

Therefore x cannot belong to I(a).


ii) (4.14) is verified also for some discontinuous functions as for in-

stance a(x) = sign(x). This confirms what was a"rmed previ-

ously, i.e. the weak existence (and uniqueness in law) for E(a, 0).

iii) If a(x) = 1{0}(x), (4.14) is not verified.

iv) If a(x) = |x|#,) $ 12 then

Z(a) = I(a) = {0}.

So there is at most one solution in law for E(a, 0).

v) The proof is technical and makes use of Levy characterisation the-

orem of Brownian motion.

4.2 Results on pathwise uniqueness

Proposition 4.13 (Yamada-Watanabe)

Let a, b : R+ ! R " R and consider again E(a, b). Suppose b globally

Lipschitz and h : R+ " R+ strictly increasing continuous such that

• h(0) = 0;

•$ "

0

1

h2(y)dy = ', )( > 0;

• |a(t, x) % a(t, y)| * h(x % y).

Then pathwise uniqueness is verified.

Remark 4.14 • In Proposition 4.13, one typical choice is h(u) =

u#, ) > 12 .

• Pathwise uniqueness for E(a, b) holds therefore if b is globally

Lipschitz and a is Holder continuous with parameter equal to 12 .


Corollary 4.15 Suppose that the assumption of Proposition 4.13 are

verified and a, b continuous with linear growth. Then E(a, b; &) admits

strong existence and pathwise uniqueness, whenever & verifies condition

(4.13).

Proof. It follows from Propositions 4.13 and 4.8 together with Propo-

sition 4.5 ii).

Remark 4.16 Suppose d = 1. Pathwise uniqueness for E(a, b) also

holds under the following assumptions.

(i) a, b are bounded, a is time-indepedent and a $ const > 0, h as in

Proposition 4.13. This result has an analogous form in the case of

space-time white noise driven SPDEs of parabolic type, as proved

by Bally, Gyongy and Pardoux in 1994.

(ii) a independent on time, b bounded and a $ const > 0; moreover

|a(x) % a(y)|2 * |f(y) % f(x)| and f is increasing and bounded.

For illustration we provide some significant examples.

Example 4.17

Xt =

$ t

0

|Xs|#dWs, t $ 0. (4.16)

We set a(x) = |x|#, 0 < ) < 1. This is equation E(a, 0) with a(x) =

|x|#. According to Engelbert-Schmidt notations, we have Z(a) = {0}.Moreover

• If ) $ 12 then I(a) = {0}.

• If ) < 12 then I(a) = 0.

Therefore according to Proposition 4.11, E(a, 0) admits weak existence.

On the other hand, if ) $ 12 ,

|x# % y#| * h(|x % y|), (4.17)


where h(z) = z#. According to Proposition 4.13, (4.16) admits pathwise

uniqueness and by Corollary 4.15, also strong existence. The unique

solution is X ( 0.

If ) < 12 , X ( 0 is always a solution. This is not the only one; even

uniqueness in law is not true.

Example 4.18 Let a(x) =!|x|, b Lipschitz.

Then E(a, b) admits strong existence and pathwise uniqueness.

In fact, a is Holder continuous with parameter 12 and the second item

of Remark 4.14 applies; so pathwise uniqueness holds. Strong existence

is a consequence of Propositions 4.8 and 4.5 ii).

An interesting particular case is provided by the following equation.

Let x0, ", ' $ 0, k # R. The following equation admits strong existence

and pathwise uniqueness.

Zt = x0 + "

$ t

0

!|Zs|dWs +

$ t

0

(' % kXs)ds, t # [0, T ]. (4.18)

Equation (4.18) is widely used in mathematical finance and it consti-

tutes the model of Cox-Ingersoll-Ross: the solution of the mentioned

equation represents the short interest rate.

Consider now the particular case where k = 0, " = 2. According to some

comparison theorem for SDEs, the solution Z is always non-negative

and therefore the absolute value may be omitted. The equation becomes

Zt = x0 + 2

$ t

0

!ZsdWs + 't. (4.19)

Definition 4.19 The unique solution Z to

Zt = x0 + 2

$ t

0

!ZsdWs + 't (4.20)

is called square '-dimensional Bessel process starting at x0; it is

denoted by BESQ$(x0); for fine properties of this process, see [17], ch.

IX.3.


Since Z $ 0, we call '-dimensional Bessel process starting from x0

the process X =1

Z. It is denoted by BES$(x0).

Remark 4.20 Let d $ 1. Let W = (W 1, · · · , W d) be a classical

d%dimensional Brownian motion. We set Xt = &Wt&. (Xt)t"0 is a

d-dimensional Bessel process.

Remark 4.21 If ' > 1, it is possible to see that

Xt = Wt +' % 1

2

$ t

0

ds

Xs.

4.3 The case with distributional drift

Pioneering work about di!usions with generalized drift was presented

by N.I. Portenko, but in the framework of semimartingale processes.

Recently some work was done caracterizing solutions in the class of the

so called Dirichlet processes, with some motivations in random irregular

environment.

An useful transformation in the theory of stochastic di!erential equa-

tion is the so called Zvonkin transformation. Let (Wt) be an (Ft)-

classical Brownian motion. Let a, b : R " R respectively C1 and locally

bounded. We suppose moreover a > 0. We fix x0 # R. Let (Xt)t"0 be

a solution of

Xt = x0 +

$ t

0

b(Xs)ds +

$ t

0

a(Xs)dWs. (4.21)

We set $(x) =

$ x

0

2b

a2(y)dy and we define h : R " R such that

h(0) = 0, h' = e$!.

h is strictly increasing. We set a(x) = (ah')(h$1(x)) where h$1 is the

inverse of h. We set Yt = h(Xt). Without entering into details, the

classical Ito formula allows to show that (Yt) defines a solution of"

dYt = a(Yt)dWt

Y0 = h(x0).(4.22)


Now, equation (4.22) fulfills the requirements of Engelbert-Schmidt cri-

terion so that it admits weak existence and uniqueness in law. Con-

sequently, unless explosion, one can easily establish the same well-

posedness for (4.21).

Zvonkin transformation also allows to prove strong existence and path-

wise uniqueness results for (4.21; for instance when

• a has linear growth;

• y "# y0

b(s)a2(s)ds is a bounded function.

In fact, problem (4.22) satisfies pathwise uniqueness and strong exis-

tence since the coe"cients are Lipschitz with linear growth. Therefore

one can deduce the same for (4.21).

Veretennikov generalized Zvonkin transformation to the d-dimensional

case in some cases which include the cas a = 1 and b bounded Borel.

Zvonkin’s procedure suggests also to consider a formal equation of the

type

dXt = dWt + #'(Xt)dt, (4.23)

where # is only a continuous function and so b = #' is a Schwartz

distribution; # could be for instance the realization of an independent

Brownian motion of W . Therefore equation (4.23) is motivated by

the study of irregular random media. When " = 1, b = #', stochastic

di!erential equation (4.22), h' = e$2% still makes sense.

Using Engelbert-Schmidt criterion, one can see that problem (4.22) still

admits weak existence and uniqueness in the sense of distribution laws.

If Y is a solution of (4.22), X = h$1(Y ) provides a natural candidate

solution for (4.21). R.F. Bass, Z-Q. Chen and F. Flandoli, F. Russo, J.

Wolf investigated generalized SDEs as (4.23): in particular they made

previous reasonning rigorous, respectively in the case of strong and

weak solutions, see [5].


5 Connected topics

We aim here at giving some basic references about topics which are

closely connected to stochastic di!erential equations.

• Stochastic partial di!erential equations (SPDEs)

If a stochastic di!erential equation is a random perturbation of

an ordinary di!erential equation, a stochastic partial di!erential

equation is a random perturbation of a partial di!erential equa-

tion. Big activity was performed in the parabolic (evolution equa-

tion) and hyperbolic case (wave equation). Most of the work was

done in the case of a fixed underlying probability spaces. We only

quote two basic monographies which should be consulted at first

before getting into the subject: the one of J. Walsh [19] and the

one of G. Da Prato and J. Zabczyk [4].

However it was possible to establish some results about weak ex-

istence and uniqueness in law for SPDEs. On possible tool was a

generalization of Girsanov theorem to the case of Gaussian space-

time white noise. Weak existence for the stochastic quantization

equation was proved with the help of infinite dimensional Dirich-

let forms by S. Albeverio and M. Rockner.

We also indicate a beautiful recent monography by G. Da Prato

which pays particular attention to Kolmogorov equations with

infinitely many variables, see [3].

• Numerical approximations

Relevant work was done in numerical approximation of solutions

to SDEs and related approximations of solutions to linear parabolic

equations via Feynmann-Kac probabilistic representation, see The-

orem 3.3). It seems that the stochastic simulations (of improved

Monte Carlo type and related topics) for solving deterministic

problems are e"cient when the space dimension is greater than

4.


• Malliavin calculus

Malliavin calculus is a wide topic which is the object of another

article in this Encyclopedia. Relevant applications of it concern

stochastic (ordinary and partial) di!erential equations. We only

quote a monography of D. Nualart on those applications, [12].

Two main objects were studied.

– Given a solution of a stochastic di!erential equation (Xt),

su"cient conditions so that the law of Xt, t > 0, has a

(smooth) density p(t, ·). Small time asymptotics of this den-

sity, when t " 0, and small drift perturbation were per-

formed, refining Freidlin-Ventsell large deviation estimates.

– Coming back to SDE (4.11), one can conceive to consider

coe"cients a, b non adapted with respect to the underlying

filtration (Ft). On the other hand, the initial condition !

may be anticipating, i.e. not F0-measureble. In that case

the Ito integral# t

0 a(s, Xs)dWs is not defined. A replacement

tool is the so called Skorohod integral.

• Rough paths approach

A very successful and significant research field is the rough path

theory. In the case of dimension d = 1, Doss-Sussmann method

allows to transform the solution of a stochastic di!erential equa-

tion into the solution of an ordinary (random) di!erential equa-

tion. In particular, that solution can be seen as depending (path-

wise) continuously from the driving Brownian motion (Wt) with

respect to the usual topology of C([0, T ]). Unless exceptions,

this continuity does not hold in case of general dimension d > 1.

Rough paths theory, introduced by T. Lyons, allows to recover

somehow this lack of continuity and establishes a true pathwise

stochastic integration.

• SDEs driven by non-semimartingales


At the present moment, there is a very intense activity towards

SDEs driven by processes which are not semimartingales. In

this perspective, we list SDEs driven by fractional Brownian mo-

tion with the help of rough paths theory, using fractional and

Young type integrals and involving finite cubic variation pro-

cesses. Among the contributors in that area we quote L. Coutin,

R. Coviello, M. Errami, M. Gubinelli, Z. Qian, F. Russo, P. Val-

lois, M. Zahle.

KEY WORDS

Stochastic ordinary (partial) di!erential equations, Feynmann-Kac for-

mula, probabilistic representation of a (linear or non-linear) PDE, Brow-

nian motion, semimartingales, di!usion process, pathwise uniqueness,

strong existence, weak existence, uniqueness in law, Peano phenom-

ena, Girsanov theorem, Bessel process, Malliavin calculus, Skorohod

integral.

SEE ALSO

63 RANDOM WALK IN RANDOM ENVIRONMENTS

137 INTERACTING STOCHASTIC PARTICLE SYSTEMS

372 RANDOM DYNAMICAL SYSTEMS

463 MALLIAVIN CALCULUS

477 STOCHASTIC HYDRODYNAMICS


References

[1] Benachour, S., Chassaing, P., Roynette, B., Vallois, P. Processus

associes l’ equation des milieux poreux. Ann. Scuola Norm. Sup.

Pisa Cl. Sci. (4) 23 (1996), no. 4, 793–832 (1997).

[2] Bouleau, N., Lepingle, D., Numerical methods for stochastic pro-

cesses. John Wiley, 1994.

[3] Da Prato, G., Kolmogorov equations for stochastic PDEs.

Birkauser, 2004.

[4] Da Prato, G., Zabczyk, J., Stochastic equations in infinite dimen-

sions. Encyclopedia of Mathematics and its Applications, 44. Cam-

bridge University Press, Cambridge, 1992.

[5] Flandoli, F., Russo, F., Wolf, J. Some stochastic di!erential equa-

tions with distributional drift. Part I: General Calculus. Osaka

Journal of Mathematics. Vol. 40, No 2, 493-542 (2003).

[6] Fleming, W.H., Soner, M. Controlled Markov processes and vis-

cosity solutions. Springer-Verlag 1993.

[7] Graham, C., Kurtz, Th. G., Meleard, S., Protter, Ph. E., Pul-

virenti, M., Talay, D. Probabilistic models for nonlinear partial dif-

ferential equations. Lectures given at the 1st Session and Summer

School held in Montecatini Terme, May 22–30, 1995. Edited by Ta-

lay and L. Tubaro. Lecture Notes in Mathematics, 1627. Springer-

Verlag. Centro Internazionale Matematico Estivo (C.I.M.E.), Flo-

rence, 1996.

[8] Kloeden, P.E., Platen, E. Numerical solutions of stochastic di!er-

ential equations. Springer-Verlag, 1992.

[9] Karatzas, I., Shreve, S.E., Brownian Motion and Stochastic Cal-

culus. Springer Verlag, Second edition, 1991.


[10] Lamberton, D., Lapeyre, B., Introduction au calcul stochastique

et applications la finance. Collection Ellipses, 1997.

[11] Ma, J., Yong, J. Forward-backward stochastic di!erential equa-

tions and their applications. Lecture Notes in Mathematics, 1702.

Springer-Verlag, 1999.

[12] Nualart, D., The Malliavin Calculus and Related Topics. Springer

Verlag, 1995.

[13] Øksendal, B., Stochastic di!erential equations, Springer-Verlag.

Fifth Edition, 1998.

[14] Pardoux, E., Backward stochastic di!erential equations and viscos-

ity solutions of systems of semilinear parabolic and elliptic PDEs

of second order. Stochastic analysis and related topics, VI (Geilo,

1996), 79–127, Progr. Probab., 42, Birkauser, 1998.

[15] Protter, P., Stochastic Integration and Di!erential Equations.


[16] Lyons, T., Qian, Z. Controlled systems and rough paths. Oxford

University Press, Oxford, 2002.

[17] Revuz, D., Yor, M., Continuous Martingales and Brownian Mo-

tion. Springer-Verlag, 1994.

[18] Stroock, D., Varadhan, Multidimensional di!usion processes.


[19] Walsh, J.B., An introduction to stochastic partial di!erential equa-

tions. Ecole d’ete de probabilites de Saint-Flour, XIV—1984, 265–

439, Lecture Notes in Math., 1180, Springer-Verlag, 1986.

[20] Yong, J., Zhou, X. Y., Stochastic controls: Hamiltonian systems

and HJB equations. Springer-Verlag, 1999.

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