qr-based methods for computing lyapunov exponents …€¦ · exponents of stochastic differential...

INTERNATIONAL JOURNAL OF c© 2010 Institute for ScientificNUMERICAL ANALYSIS AND MODELING, SERIES B Computing and InformationVolume 1, Number 2, Pages 147–171

QR-BASED METHODS FOR COMPUTING LYAPUNOV

EXPONENTS OF STOCHASTIC DIFFERENTIAL EQUATIONS

FELIX CARBONELL, ROLANDO BISCAY, AND JUAN CARLOS JIMENEZ

Abstract. Lyapunov exponents (LEs) play a central role in the study of stability properties andasymptotic behavior of dynamical systems. However, explicit formulas for them can be derived forvery few systems, therefore numerical methods are required. Such is the case of random dynamicalsystems described by stochastic differential equations (SDEs), for which there have been reportedjust a few numerical methods. The first attempts were restricted to linear equations, which haveobvious limitations from the applications point of view. A more successful approach deals withnonlinear equation defined over manifolds but is effective for the computation of only the top LE.In this paper, two numerical methods for the efficient computation of all LEs of nonlinear SDEsare introduced. They are, essentially, a generalization to the stochastic case of the well knownQR-based methods developed for ordinary differential equations. Specifically, a discrete and acontinuous QR method are derived by combining the basic ideas of the deterministic QR methodswith the classical rules of the differential calculus for the Stratanovich representation of SDEs.Additionally, bounds for the approximation errors are given and the performance of the methodsis illustrated by means of numerical simulations

Key words. Lyapunov Exponents, Stochastic Differential Equations, QR-decomposition, numer-ical methods.

1. Introduction

Since A.M. Lyapunov introduced the concept of characteristic exponents [27], ithas played an important role in the study of the asymptotic behavior of dynamicalsystems. In particular, the Lyapunov exponents (LEs) have been extensively usedfor analyzing stability properties of dynamical systems [1]. Some other importantcontributions like [4] and [5] extend the classical theory of LEs from deterministicdynamical systems to Random Dynamical Systems (RDS). This has allowed thestability analysis of a wide class of physical, mechanical and engineering processes,where the randomness becomes an essential issue for modeling their dynamics [32],[3], [36], [39], [30], [20], [34], [26], [9].

It is well known that, with the exception of some simple cases as those mentionedin [2], explicit formulas for the LEs of Stochastic Differential Equations (SDEs) arerarely known. Alternatively, some different analytic expansions have been obtainedfor equations driven by particular sources of noise [37]. On the other hand, as-ymptotic expansions in terms of noise intensity have been obtained for LEs oftwo-dimensional equations driven by a small noise [6], [33]. Some other asymptoticexpansions for LEs have been also obtained in [3] for more general cases of large,small and slow noises. Other types of parametric expansions have been reported forcertain particular systems [22], [23], [8]. In principle, the truncation of any of suchexpansions could be used as a numerical method for computing LEs. However thisprocedure lacks generality since it is only applicable for certain particular cases ofSDEs.

Received by the editors May 30, 2010 and, in revised form, November 30, 2010.2000 Mathematics Subject Classification. 34D08, 37M25, 60H35.This research was supported by the research grant 03-059 RG/MATHS/LA from the Third

World Academy of Science.

147

148 FELIX CARBONELL, ROLANDO BISCAY, AND JUAN CARLOS JIMENEZ

Indeed, the development of numerical methods for computing LEs of generalSDEs is a relevant issue that has not received a systematic attention. In fact,just a few papers have addressed such subject with a relative success [36], [19],[38]. The seminal works in this respect are the numerical method proposed in[36] and [38] for the class of linear stochastic equations. Afterward, this methodwas extended to the general case of nonlinear SDEs defined either on a compactorientable manifold or on R

d [19]. It was based on the discretization of the SDEby particular integrators and the corresponding approximation of the LEs for theresulting ergodic Markov chains. From a practical point of view, this method is justeffective for the computation of the top LE, but it leads to numerical instabilitiesfor the remaining ones [36].

The aim of this paper is to introduce two alternative methods for the numericalcomputation of the LEs of nonlinear SDEs defined on R

d. These methods are,essentially, a generalization to the stochastic case of the well known QR-basedmethods for the computation of the LEs of Ordinary Differential Equations (ODEs)[10], [11], [18], [16], [17]. Specifically, a discrete and a continuous QR methodare obtained by combining the basic ideas of the deterministic methods with theclassical rules of the differential calculus for the Stratanovich representation ofSDEs. In particular, the discrete QR method generalizes the algorithm presentedin [19] for the top LE. Moreover, in contrast with that algorithm, the methodsintroduced here allow the efficient computation of all LEs.

The outline of the paper is as follows. Basic notations as well as essential factsabout LEs of SDEs are presented in Section 2. In Section 3 the discrete andcontinuous QR methods are derived, whereas bounds for the approximation errorsare given in Section 4. In section 5, some suggestions for the implementation ofthe QR methods are presented. Finally, in the last section, the performance of themethods is illustrated by means of simulations.

2. Preliminaries

2.1. Notations. For any matrix A, let us denote by Ak, Akl, A|k and A|kk thek− th column vector, the entry (k, l), the matrix of first k columns and the matrixof the first k rows and columns of A, respectively. The Frobenius norm for matricesshall be denoted by ‖·‖ , and the standard scalar product in R

d by < ·, · >. For

any k ∈ Z+ and 0 ≤ δ ≤ 1 let Ck,δ

b be the Banach space of Ck vector fields on Rd

with growth at most linearly and bounded derivatives of order 1 up to k, and whosek − th derivatives are globally δ-Holder continuous. Finally, denote by Cl

P (Rd,R)

the space of l time continuously differentiable functions g : Rd → R for which g andall its partial derivatives up to order l have polynomial growth.

2.2. Lyapunov Exponents of Nonlinear SDEs. Let Ft, t ≥ 0 be an increas-ing right continuous family of complete sub σ-algebras of F and let (Ω,F ,P) bethe canonical Wiener space on R

+ (i.e., Ω = ω ∈ C(R,Rm) : ω(0) = 0, F = B(Ω)is the Borel σ-algebra of Ω, and P is the Wiener measure in F).

Consider the Stratanovich SDE on Rd,

(1) dxt =

m∑

i=0

fi (xt) dwit, xto = x0, t ≥ t0 ≥ 0,

and the variational equation on Rd× R

d,

COMPUTING LYAPUNOV EXPONENTS OF SDES 149

(2) dVt =

m∑

i=0

Dfi (xt)Vt dwit, Vt0 = Id, t ≥ t0 ≥ 0,

where w = (w1, .., wm) is an m-dimensional Ft-adapted standard Wiener process,with the convention dw0

t = dt. Here, it is assumed that the functions f0, f1, ..., fmsatisfy

(3) f0 ∈ Ck,δb , f1, ..., fm ∈ Ck+1,δ

b ,m∑

i=1

d∑

j=1

fji

∂

∂xjfi ∈ Ck,δ

b

for some k ≥ 1 and δ > 0; and f0 is such that the hypothesis:

(4) ∃β > 0 and K ⊂Rd compact, s.t. < x, f0(x) >≤ −β ‖x‖2 , ∀x ∈ R

d\Kholds. Further, the infinitesimal generator of the process xt that solves (1),

L = f0 +1

2

m∑

i=1

f2i

is assumed to be strongly hypoelliptic in the sense that dimL(f0, f1, ..., fm)(x) =d for all x ∈ R

d, where L(f0, f1, ..., fm) denotes the Lie algebra generated by thevector fields f0, ..., fm.

Denote by (Ω,F ,P , (θt)t∈R) the ergodic metric dynamical system defined on(Ω,F ,P), where

θtω(·) = ω(t+ ·)− ω(t), t ∈ R

is the measuring preserving and ergodic shift transformation (see appendices in [5]for details). It is well known (Theorem 2.3.32 in [5]) that equation (1) generates aCk RDS (or cocycle) ϕ : R × Ω × R

d → Rd over θ. Therefore, the system (1)-(2)

generates the Ck−1 RDS (ϕ,Dϕ) over θ, which means thatDϕ solves the variationalequation (2) and is a linear matrix cocycle over the skew-product (metric dynamicalsystem) Θ = (θ, ϕ).

In addition, let µ be an ergodic invariant measure with respect to the cocycle ϕ,

i.e., a ergodic measure on (Ω×Rd,F ⊗ B(Rd

)) that satisfy Θ(t)µ = µ for all t ≥ 0and πΩµ = P , where πΩ denotes the projection onto Ω (see Remark below). Thus,the multiplicative ergodic theorem (MET) of Oseledets ([31]) for linear cocyclesasserts the regularity and the existence of the Lyapunov spectrum (see also pages 1-26 in [4] for an excellent review). In particular, by Theorem 4.2.13 in [5], there exits

an invariant set Ω ∈ F ⊗ B(Rd) of full µ-measure, constants λ1 ≥ λ2 ≥ ... ≥ λp,

p ≤ d (the LEs of the system (1)-(2)), numbers di ∈ N withp∑

i=1

di = d (their

multiplicities), and a random splitting

Rd = E1(ω)⊕ ...⊕ Ep(ω)

of Rd into measurable random subspaces Ei(ω), ω ∈ Ω (the so-called Oseledets

splitting) with dim(Ei(ω)) = di such that

λj = limt→∞

1

tlog ‖Vtpj‖ if and only if pj ∈ Ej(ω)\0.

Here, the set of vectors p1, ..,pd is an orthonormal basis of Rd, which is usuallycalled the Lyapunov normal basis. Moreover, the Lyapunov regularity condition


can be expressed as

(5)

d∑

j=1

λj = lim supt→∞

1

tlog |det(Vt)| .

Remark According to Theorem 2.3.45 in [5], µ = P × ρ is an invariant measurerespecting to ϕ if and only if ρ is a stationary measure of the Markov transitionprobability of the one-point motion of ϕ. In particular, µ is ergodic if and only if ρis ergodic. Thus, to fulfill the requirements of the Multiplicative Ergodic Theoremit is enough to choose an ergodic measure ρ and then define µ := P × ρ. To dothis just recall that the one-point motion of ϕ has a unique invariant probabilitymeasure ρ with positive density ν respecting to the Lebesgue measure on R

d if, forinstance, the infinitesimal generator L is elliptic, or more generally if L is stronglyhypoelliptic [21] and condition (4) holds [19].

3. QR methods for computing LEs

In this section the discrete and the continuous QR method for computing theLEs of (1)-(2) are derived. Basically, the same formulation of [16] and [17] for theQR methods in ODEs is followed here. Thus, the continuous QR factorization ofVt shall be the main tool for our computational purposes.

3.1. QR formulation for LEs. Let us rewrite the equation (2) as

dVt =m∑

i=0

Ai (t)Vt dwit, Vt0 = Id,

where Aj(t) = Dfj (xt) , and consider the continuous QR factorization of Vt,

Vt = QtRt,

where Qt is orthogonal and Rt is upper triangular with positive diagonal elementsR

jjt , j = 1, ..., d. Hence, by the norm-preserving property of Qt it is obtained

λj = limt→∞

1

tlog ‖Vtpj‖ = lim

t→∞

1

tlog ‖Rtpj‖ ,

where pj, j = 1, ..., d denotes any orthonormal basis associated to the splittingof Rd.

Moreover, for each 1 ≤ k ≤ d, the solution matrix V|kt may be considered

as a diffusion process in the Grassmann bundle Gk(Rd) (the manifold of the k-

dimensional subspaces of Rd). Then, for each t, the matrix V|kt may be seen as

the direction of Vte1 ∧Vte2 ∧ ... ∧Vtek in the kth exterior power Λk(Rd), whereeii=1...d is the standard basis of Rd. Hence, according to corollary 2.2 in [7],

λ1 + λ2 + ...+ λk = limt→∞

1

tlog ‖Vte1 ∧Vte2 ∧ ... ∧Vtek‖

= limt→∞

1

tlog

√det((< Vtei,Vtej >)i,j=1,...,k)

= limt→∞

1

tlog(

√det((V

|kt )⊤V

|kt )) a.s.


Since V|kt = Q

|kt R

|kkt then,

λ1 + λ2 + ...+ λk = limt→∞

1

tlog

∣∣∣det((R|kkt )⊤R

|kkt )

12

∣∣∣ = limt→∞

1

tlog

∣∣∣det(R|kkt )

∣∣∣

= limt→∞

1

tlog

(Πk

i=1Riit

)= lim

t→∞

1

t

k∑

i=1

logRiit a.s,

which implies

(6) λj = limt→∞

1

tlog

∣∣∣Rjjt

∣∣∣ a.s.

In practice, the expression above should be approximated at a finite instant oftime T ≥ t0. This leads to the following definition.

Definition 3.1. For a given time t0 ≤ T <∞, the truncated LEs at T are definedby

λj (T ) =1

T − t0log

∣∣∣RjjT

∣∣∣

3.2. Discrete QR method. Let

(t)h = t0 ≤ t1 ≤ ... ≤ tn < ... <∞be a time partition defined as a sequence of Ftn -measurable random times tn,n = 0, 1, .., that satisfy

supn(tn+1 − tn) ≤ h < 1, w.p.1,

and definent := maxn = 0, 1, 2, ..., : tn ≤ t <∞.

The main idea of the discrete QR method is to indirectly compute the funda-mental solution matrix of the equation (2) by successive QR factorization of Vt atdiscrete times tn ∈ (t)h, and then, expressing the LEs in terms of the factors Rt.

Algorithm DescriptionSet

Vt0 = Qt0 = Id.

For s = 0, 1, ... and any initial condition x0 solve the system

(7) dxt =m∑

i=0

fi (xt) dwit, xt0 = x0 ts ≤ t ≤ ts+1

(8) dZt =

m∑

i=0

Dfi (xt)Zt dwit, Zts = Qts ts ≤ t ≤ ts+1

and then take the QR factorization

Zts+1= Qts+1

Rts+1.

Then one has

Vts+1= Zts+1

Q⊤tsVts = Qts+1

Rts+1Q⊤

tsVts = ... = Qts+1Rts+1

...Rt1

Hence, the LEs and the truncated LEs are obtained by

λj = lims→∞

1

ts

s∑

n=1

log∣∣∣Rjj

tn

∣∣∣


and

(9) λj(T ) =1

T − t0

nT∑

n=1

log∣∣∣Rjj

tn

∣∣∣ ,

respectively.

Remark It should be noted that the method proposed in [19] for computing thetop LE is a particular case of the algorithm above. Indeed, the QR decomposition

of Zt gives R11t =

∥∥Z1t

∥∥ and Q1t =

Z1t

‖Z1t‖, and hence,

λ1 = lims→∞

1

ts

s∑

n=1

log∥∥Z1

tn

∥∥ ,

where Z1t is the first column vector of the matrix Zt, which in turn is solution of

dyt =

m∑

i=0

Dfi (xt)yt dwit, yts =

Z1ts∥∥Z1ts

∥∥ ts ≤ t ≤ ts+1.

Then, the two previous expressions leads to the algorithm stated in [19]. Moreover,the approximation of all the LEs in [19] requires the computation of

1

Tlog(

√det((V

|kT )⊤V

|kT ))

to get successively the (approximated) sums λ1 +λ2+ ...+λk, which is much moreinefficient (and perhaps unstable) than the discrete QR algorithm presented in thissection. Finally, note also that in the case of null stochastic components in theequations (1)-(2), the proposed algorithm reduces to the conventional QR discretemethod for ODEs in [16] and [17].

3.3. Continuous QR method. The general idea of the continuous QR methodis to obtain an SDE for the factor Q and then evaluating a Lebesgue integral foreach of the LEs.

By taking differentials in the sense of Stratanovich in the equalities Vt = QtRt

and Q⊤t Qt = Id it is obtained

dVt = (dQt)Rt +Qt (dRt)(10)

0 =(dQ⊤

t

)Qt +Q⊤

t (dQt) .(11)

Since

dVt =m∑

i=0

Dfi (xt)QtRt dwit

holds, it is deduced from (10) that

(12) (dRt)R−1t =

m∑

i=0

Q⊤t Dfi (xt)Qt dwi

t − dSt,

where

dSt = Q⊤t (dQt) .

Whereas, from the equality (11) it is concluded that dSt is a skew-symmetric matrix.


Moreover, given that (dRt)R−1t is an upper triangular matrix it is deduced from

(12) that St satisfies the equation

(13) dSjlt =

m∑i=0

(Q⊤

t Dfi (xt)Qt

)jl dwit j > l

0 j = l

−m∑i=0

(Q⊤

t Dfi (xt)Qt

)jl dwit j < l

,

which implies the following SDE for Qt :

(14) dQt = QtdSt =

m∑

i=0

QtTit (xt,Qt) dwi

t,

where the matrices Tit (xt,Qt) , i = 0, ...,m are given by

(Ti

t(xt,Qt))jl

=

(Q⊤

t Dfi (xt)Qt

)jlj > l

0 j = l

−(Q⊤

t Dfi (xt)Qt

)jlj < l

, j, l = 1, ..., d

From (12) and (13) it is obtained

(15) dRt =

m∑

i=0

(Q⊤t Dfi (xt)Qt −Ti

t (xt,Qt))Rt dwit,

which implies the following equation for each diagonal element Rjjt , j = 1, ..., d :

(16) dRjjt =

m∑

i=0

(Q⊤t Dfj (xt)Qt)

jjRjjt dwi

t

Therefore,

(17) λj = limt→∞

1

tlogRjj

t = limt→∞

1

t

m∑

i=0

t∫

t0

(Q⊤uDfi (xu)Qu)

jj dwiu.

Note that this formula is a straight generalization to the stochastic case of thecontinuous QR method proposed in [16] and [17]. However, this expression involvesthe computation of m Stratanovich integrals, which could be too expensive from apractical point of view. Hence, an alternative expression is in order.

From equation (16) it is deduced that

d(det(R|jjt )) =

m∑

i=0

tr((Q⊤t Dfi (xt)Qt)

|jj) det(R|jjt ) dwi

t

=

m∑

i=0

tr(Dfi (xt)Q|jt (Q

|jt )

⊤) det(R|jjt ) dwi

t,

for each 1 ≤ j ≤ d. Now, let Pjt be the orthogonal projection on the j−dimensional

subspace of Rd spanned by the first j column vectors of Vt, that is

Pjt = V

|jt ((V

|jt )

⊤V|jt )

−1(V|jt )

⊤.

Using that V|jt = Q

|jt R

|jjt ,such orthogonal projector can be rewritten as

Pjt = Q

|jt (Q

|jt )

⊤,


which implies that

d(det(R|jjt )) =

m∑

i=0

tr(Dfi (xt)Pjt ) det(R

|jjt ) dwi

t.

Then, by using the Ito formula for log(det(R|jjt )) it is obtained the following equa-

tion in Ito form (see details in Theorem 3.1 in [7]):

d log(det(R|jjt )) = ψj (xt,Qt) dt+

m∑

i=1

tr(Dfi (xt)Q|jt (Q

|jt )

⊤)dwit,

where

ψj (x,Q) = tr(Df0 (x)Q|jt (Q

|jt )

⊤) +1

2

m∑

i=1

[tr(D (Dfi (x) fi (x))Q|jt (Q

|jt )

⊤)

+tr((Dfi (x))⊤ (Id −Q|j(Q|j)⊤)Dfi (x)Q

|jt (Q

|jt )

⊤)]

−(tr(Dfi (x)Q|jt (Q

|jt )

⊤))2.(18)

Therefore, as it was shown in Corollary 3.1 of [7],

limt→∞

1

t

t∫

0

m∑

i=1

tr(Dfi (xt)Pjt )dw

it = 0 a.s,

which yields

(19) λ1 + λ2 + ...+ λj = limt→∞

1

tlog(det(R

|jjt )) = lim

t→∞

1

t

t∫

0

ψj (xs,Qs) ds a.s.

Hence,

λj = limt→∞

1

t

t∫

0

∆ψj (xs,Qs) ds a.s.,

for all j = 1, ..., d, where ∆ψj = ψj − ψj−1, with ψ0 ≡ 0.Note that the previous expression involves the computation of a single Riemann

integral, which is easier to evaluate than expression (17).Algorithm DescriptionThe continuous QR algorithm is based on the integration of the system of equa-

tions

(20) dxt =

m∑

i=0

fi (xt) dwit, xt0 = x0

(21) dQt =

m∑

i=0

QtTit (xt,Qt) dwi

t, Qt0 = Id

to obtain the factor Q at the points t0 < t1 < .... < ts < ... and then computingthe LEs by

λj = lims→∞

1

ts

s−1∑

n=0

tn+1∫

tn

∆ψj (xu,Qu) du.


Remark The matrix Qt may be considered as a diffusion process that induce aRDS on the projective bundle P (Λd(Rd))(see Section 6.4.2 in [5] for more details).Hence, the existence of an unique invariant ergodic measure for such RDS and theergodicity of (xt,Qt) is guarantied by Theorem 6.4.5 in [5]. It is worth noting thatfor the derivation of (19) we basically followed the Baxendale’ results [7] (or moregenerally Section 6.4.2 in [5]). Indeed, the expression (19) is closely related withthe Furstenberg-Khasminskii formula for the sum of LEs.

3.4. Numerical Approximations. It is well-known that, in the framework ofSDEs, the use of weak integrators is the most appropriate choice for the approxi-mation of LEs [35], [36], [19], [25]. Hence, in the sequel, it will be assumed that thesolution of the systems (7-8) and (20-21) are numerically approximated by weak

integrators, whose solutions shall be denoted by (xt, Zt) and (xt, Qt), t ∈ (t)h ,

respectively. Hence, the appoximated LEs λj , j = 1, ..., d are given by

(22) λj = lims→∞

1

ts

s∑

n=1

log∣∣∣Rjj

tn

∣∣∣ ,

in the discrete case and by

(23) λj = lims→∞

1

ts

s−1∑

n=0

tn+1∫

tn

∆ψj(xu, Qu)du.

in the continuous case. Correspondingly, the respective truncated approximationsare given by

(24) λj (T ) =1

T − t0

nT∑

n=1

log∣∣∣Rjj

tn

∣∣∣

and

(25) λj(T ) =1

T − t0

nT−1∑

n=0

tn+1∫

tn

∆ψj(xu, Qu)du,

where, without loss of generality it is assumed that tnT = T.

It should be noted that, in general, λj are not the LEs of the discrete Markov

chains (xt, Zt) and (xt, Qt), t ∈ (t)h that approximate the continuous ergodic pro-cess (xt,Zt) and (xt,Qt), respectively. In [36] and [19] the authors demonstrate theapplicability of a discrete Multivariate Ergodic Theorem and as a consequence theexistence of LEs for the Markov chains resulting from particular numerical integra-tors, namely, the weak Euler and Milstein schemes. As one can see there, provingthe existence of LEs for these particular schemes requires a tedious and long al-gebraic manipulations, mainly addressed to showing that the ergodicity property

of the process (xt,Zt) and (xt,Qt) is inherited by the Markov chains (xt, Zt) and

(xt, Qt), t ∈ (t)h. Our approach here differs from that in [36] and [19] in the sense

that we are not assuming that λj are LEs but they are just numerical approxima-tions to the true LEs λj . However, our numerical simulations carried out in thelast section were implemented with both weak Milstein and weak Euler schemes,for which the ergodicity property has been already proved.


4. Convergence Analysis

In this section an analysis of the approximation errors in the computation ofthe LEs is carried out. The results presented here are somehow quite similar tothose given in [17] and [28] for the case of the QR methods in ODEs. However, a

difference appears in the stochastic case, namely, the quantities λj (T ) are randomvariables. Since the LEs are in fact functionals of the solution process of (1)-(2), then the weak convergence criterion is the more natural choice to deal with[36], [19]. However, the approach followed in this section is somewhat differentfrom that of [36], [19]. There, the authors obtained convergence results for theparticular case of the Euler and Milstein schemes when T goes to infinity, provided

the approximations λj were in fact LEs of discrete Markov chains (resulting formthe numerical schemes). Our results here corresponds to the standard convergenceanalysis carried out when dealing with weak approximations, computed as usual,up to a finite time instant. Specifically, our approach consists on giving bounds for

the errors∣∣∣λj − E(λj(T ))

∣∣∣ , i = 1, ..., d, which can be decomposed as

∣∣∣λj − E(λj(T ))∣∣∣ ≤ |λj − E(λj(T ))|+

∣∣∣E(λj(T ))− E(λj(T ))∣∣∣ .

From this perspective, a convergence analysis for both terms |λj − E(λj(T ))| and∣∣∣E(λj(T ))− E(λj(T ))∣∣∣ will be given in the sequel. For the first term we have the

following theorem, which is a stochastic version of Lemma 4.2 in [17].

Theorem 4.1. Let T > 0 be a large enough fixed number. Then, for each ǫ > 0there exists a positive constant Kǫ independent of T such that

|λj − E(λj (T ))| ≤Kǫ

T − t0+ ǫ

Proof. Denote by Rǫ : Ω → [1,∞) an ǫ−slowly random variable defined as in [5].That is,

(26) −ǫ+ 1

tlog (Rǫ) ≤

1

tlog (Rǫ(θt)) ≤

1

tlog (Rǫ) + ǫ,

where θt is the ergodic shift transformation of the ergodic metric dynamical system(Ω,F ,P , (θt)t∈R) defined in Section 2.

According to Theorem 4.3.4 in [5], for each ǫ > 0 there exists an ǫ−slowly randomvariable Rǫ such that

1

Rǫeλjt−ǫ|t| ‖x‖ ≤ ‖Vtx‖ ≤ Rǫe

λjt+ǫ|t| ‖x‖ ,

where x is any vector in the subspace of Rd corresponding to the LE λj accordingto the Lyapunov spectrum of (1)-(2). Taking ‖x‖ = 1 and t = T − t0 it is obtained

−ǫ− 1

T − t0log (Rǫ) ≤ λj (T )− λj ≤

1

T − t0log (Rǫ) + ǫ,

which implies

(27) |λj − E(λj (T ))| ≤1

T − t0E(|log (Rǫ)|) + ǫ.

On the other hand, (26) implies that

−ǫ ≤ lim inft→∞

1

tlog (Rǫ(θt)) ≤ lim sup

t→∞

1

tlog (Rǫ(θt)) ≤ ǫ,


and so

E(lim supt→∞

1

t|log (Rǫ(θt))|) ≤ ǫ.

Thus, by the Fatou’s Lemma,

lim supt→∞

E(1

t|log (Rǫ(θt))|) ≤ ǫ.

Therefore, for all ǫ′ > 0 there exists a T ′ such that

(28) E(1

t|log (Rǫ(θt))|) ≤ ǫ

′′

for all t > T ′, where ǫ′′

= ǫ+ ǫ′. Additionally, (26) also implies that

−ǫ+ 1

tlog (Rǫ(θt)) ≤

1

tlog (Rǫ) ≤

1

tlog (Rǫ(θt)) + ǫ

for all t. Thus, ∣∣∣∣1

tlog (Rǫ)

∣∣∣∣ ≤∣∣∣∣1

tlog (Rǫ(θt))

∣∣∣∣+ ǫ.

and so1

tE(|log (Rǫ)|) ≤

1

tE(|log (Rǫ(θt))|) + ǫ.

This and (28) imply that

Kǫ = E(|log (Rǫ)|) <∞.

With this bound and inequality (27), the prove is concluded.

Bounds for the second error term∣∣∣E(λj(T ))− E(λj(T ))

∣∣∣ will be given in the

following two subsections.

4.1. Weak convergence to the truncated LEs: continuous QR method.

Let

yt= (x1t , ...,x

dt ,Q

11t , ...,Q

ddt ) ∈ R

d+d2

, t ∈ [t0, T ]

be the solution of the system

dxt =

m∑

i=0

fi (xt) dwit, xt0 = x0,

dQt =

m∑

i=0

QtTit (xt,Qt) dwi

t, Qt0 = Id,

and let yt= (x1t , ..., x

dt , Q

11t , ..., Q

ddt ), t ∈ (t)h be any numerical approximation to

yt. Denote by yt(s, r) the solution at t of the system above with initial conditionys = r, and let yt(s, r) be an approximation to yt(s, z) that satisfy ys = r. Hence,with this notation, it is obvious that yt(t0,yt0) = yt and yt(t0,yt0) = yt. Forl = 1, 2, ..., define Pl = p ∈ 1, ..., d+ d2l and for each p = (p1, ..., pl) ∈Pl define

the function Fp : Rd+d2 → R as

Fp(y) =

l∏

i=1

ypi ,

for all y = (y1, ...,yd+d2

). The following theorem provides condition under which aweak order β numerical approximation yt yields to a weak order β approximation

of λj (T ) to λj (T ) as well. It is obtained as a direct application of the method for


evaluating functional integrals by weak approximations (see [35] and Chapter 17 in[25]).

Theorem 4.2. Let β ∈ 1, 2, ... be given and suppose that f0, ..., fm, k = 1, .., dare such that(29)

f0 ∈ C2β+3P (Rd,R), f1, ..., fm ∈ C2β+4

P (Rd,R),

m∑

i=1

d∑

j=1

fji

∂

∂xjfi ∈ C2β+3

P (Rd,R).

In addition suppose that for each q = 1, 2, ..., there exist constants C < ∞ andr ∈ N

+, which do not depend on h, such that

(30) E( max0≤n≤nT

‖ytn‖2q /Ft0) ≤ C(1 + ‖y0‖2r),

and

(31) E(∥∥ytn+1

(tn, ytn)− ytn

∥∥2q/Ftn) ≤ C(1 + ‖ytn‖2r )(tn+1 − tn)

q,

(32)∣∣E(Fp(ytn+1(tn, ytn)− ytn)− Fp(ytn+1

(tn, ytn)− ytn)/Ftn)∣∣ ≤ C(1+‖ytn‖2r)(tn+1−tn)hβ ,

for all n = 0, ..., nT − 1, p ∈ Pl and l = 1, ..., 2β + 1. Then, there exist positive

constants Kj(T ), which do not depend on h such that∣∣∣E(λj (T ))− E(λj (T ))

∣∣∣ ≤ Kj(T )hβ,

for all j = 1, ..., d, where λj (T ) is the corresponding approximation to the truncatedLE (25).

Proof. Since

λj (T ) =1

T − t0

T∫

t0

∆ψj (xt,Qt) dt,

which implies that∣∣∣E(λj (T ))− E(λj (T ))

∣∣∣ ≤ supt∈[t0,T ]

|E(bj(yt))− E(bj(yt))| ,

where

bj(y) := ∆ψj (x,Q) ∈ C2β+2P (Rd+d2

,R)

From this and conditions (30), (31) and (32) it can be obtained from Theorem 9.1in [29] that

supt∈[t0,T ]

|E(bj(yt))− E(bi(yt))| ≤ Cj(1 + ‖y0‖2r)hβ ,

for some positive constants Cj and r ∈ N+, which concludes the proof.

4.2. Weak convergence to the truncated LEs: discrete QR method. Sim-ilarly to the previous subsection, let

yt= (x1t , ...,x

dt ,Z

11t , ...,Z

ddt ) ∈ R

d+d2

be the solution of the system (7)-(8) and let yt= (x1t , ..., x

dt , Z

11t , ..., Z

ddt ) be its

approximation. Then the following theorem holds.


Theorem 4.3. Let β ∈ 1, 2, ... be given and suppose that the components fk0 , ..., fkm,

k = 1, .., d are such that(33)

fk0 ∈ C2β+3P (Rd,R), fk1 , ..., f

km ∈ C2β+4

P (Rd,R),

m∑

i=1

d∑

j=1

fji

∂

∂xjfi ∈ C2β+3

P (Rd,R).

In addition suppose that for each q = 1, 2, ..., there exits constants C < ∞ andr ∈ N

+, which do not depend on h, such that


‖ytn‖2q /Ft0) ≤ C(1 + ‖y0‖2r),


∥∥∥Z−1tn

∥∥∥q

/Ft0) ≤ C(1 + ‖Z0‖2r),

andE(

∥∥ytn+1(tn, ytn)− ytn

∥∥2q/Ftn) ≤ C(1 + ‖ytn‖2r )(tn+1 − tn)

q,∣∣E(Fp(ytn+1

(tn, ytn)− ytn)− Fp(ytn+1(tn, ytn)− ytn)/Ftn)

∣∣ ≤ C(1+‖ytn‖2r)(tn+1−tn)hβ ,for all n = 0, ..., nT − 1, p ∈ Pl and l = 1, ..., 2β+1. Then for each g ∈ C2β+2

P there

exits positive constants Kj(T ), which do not depend on h such that∣∣∣E(λj (T ))− E(λj (T ))

∣∣∣ ≤ Kj(T )hβ,

for all j = 1, ..., d, where λj (T ) is the corresponding approximation to truncated LE(24).

Proof. According to the expressions (9) and (24) it is obtained

|E(λj (T ))− E(λj (T ))| =1

T − t0|E(log |Rjj

T |)− E(log |RjjT |)|.

The QR decomposition of the matrices ZT and ZT allows one to rewrite this ex-pression as

|E(λj (T ))− E(λj (T ))| = |E(bj (yT ))− E(bj(yT ))|,where for any

y = (x1, ...,xd,Z11, ...,Zdd) ∈ Rd+d2

.

the functions bj, j = 1, ..., d are defined by

b1(y) = log∥∥Z1

∥∥ ,

bj(y) = log

∥∥∥∥∥Zj −

j−1∑

i=1

⟨Zj ,Qi

⟩Qi

∥∥∥∥∥ ,

with

Q1 =Z1

R11, R11 =

∥∥Z1∥∥ ,

Qj =

Zj −j−1∑i=1

⟨Zj ,Qi

⟩Qi

Rjj, Rjj =

∥∥∥∥∥Zj −

j−1∑

i=1

⟨Zj ,Qi

⟩Qi

∥∥∥∥∥ , j = 2, ..., d.

It is easy to check from (33) that, for all j = 1, ..., d,

(36) bj ∈ C2β+2(Rd+d2

,R).


On the other hand, for each k = 1, ..., d+ d2,∣∣∣∣∂b1(y)

∂yk

∣∣∣∣ =∥∥Z1

∥∥−1

∣∣∣∣∣∂∥∥Z1

∥∥∂yk

∣∣∣∣∣

=∥∥Z1

∥∥−3/2∣∣∣∣< Z1,

∂Z1

∂yk>

∣∣∣∣

≤∣∣R11

∣∣−3/2 ∥∥Z1∥∥∥∥∥∥∂Z1

∂yk

∥∥∥∥

≤∣∣R11

∣∣−3+ ‖y‖2

≤∥∥Z−1

∥∥3 + ‖y‖2

In a similar way, it can be shown that for each p ∈ Pl, l = 1, ..., 2β+2 and j = 1, ..., dthere exits constants C > 0 and rpk

, spk= 1, 2, ..., such that

∣∣∣∣∂pbj(y)

∂yp

∣∣∣∣ ≤ C(

|p|∑

p=1

j∑

k=1

∣∣Rjj∣∣−rpk + ‖y‖spk )(37)

≤ C(

|p|∑

p=1

j∑

k=1

∥∥Z−1∥∥rpk + ‖y‖spk ),

If the condition bj ∈ C2β+2P (Rd+d2

,R) were true for the functions bj then it wereenough to apply Theorem 9.1 in [29] to conclude our proof. Although this is notthe case, it can be seen that, for the purpose of Theorem 9.1 in [29], conditions

bj ∈ C2β+2P (Rd+d2

,R) and (34) are equivalent to conditions (36), (37), (35) and(34). Thus, a slightly variation in the proof of such theorem yields

|E(bj(yT ))− E(bj(yT ))| ≤ Kj(T )hβ,

for certain positive constants Kj(T ), j = 1, ..., d.

It should be noted here that due to inequality (37), even with accurate approxi-mations to Zt, some difficulties can be expected for the approximation of negativeLEs of large magnitude. In fact, this has been reported in the literature regardingthe computation of LEs of ODEs by the discrete QR method.

Remark As we already mentioned, the analysis carried out in this section doesnot assume any ergodicity property on the discrete Markov chains resulting fromthe numerical approximations. This means that, in principle, our analysis can beapplied to any numerical integrator. However, for those integrators where suchergodicity property had already been proved (e.g. Euler, Milstein and the secondorder schemes given in [35]), one is able to obtain more accurate results. Indeed,ergodicity property would imply the uniform bounds (in T ) on the constantsKj(T ),j = 1, ..., d. On the other hand, under ergodicity assumptions one may obtains

bounds for the almost sure convergence of λj(or λj(T )) to λj (as in [36] and [19]).Therefore, as a general rule, we recommend the use of ergodic numerical schemesfor the approximations of LEs by QR-based methods.

5. Computational Aspects

In this section some computational aspects to take into account when applyingboth the discrete and the continuous QR algorithms are given. It is also brieflypresented a practical criterion for the choice of the truncation time T.


According to Section 3.2, the discrete QR algorithm requires the numerical in-tegration of the equations (7)-(8), which can be carry out by any of the standardweak numerical integrators for SDE [25]. However, the continuous QR algorithmrequires the integration of the system (20)-(21) in such a way of preserving theorthogonality of the factor Qt for each t ≥ t0. This can be achieved by two differentclasses of numerical schemes. The first class are usually called projected orthogonalschemes. They are based on generating approximations by any standard numericalscheme and then projecting the solution into the set of orthogonal matrices. Theschemes belonging to the second class are those that preserve the orthogonality dur-ing the numerical integration process. For that reason they are called automaticorthogonal integrators. In a recent paper [14], a class of such automatic orthogo-nal schemes for SDEs was introduced. These schemes have order of convergence1 and constitute a stochastic version of the automatic orthogonal integrators forODEs [15]. So, the continuous QR presented here can be implemented by eithersuch automatic orthogonal schemes or by a projected method, which shall be calledContinuous Automatic and Continuous Projected QR methods, respectively.

In general, as in any numerical integration problem, there is a number of practicalaspects that could improve the efficiency and accuracy of the numerical solutions of(1)-(2). Those aspects focus, for instance, on an adequate selection of the numericalintegrator and efficient adaptive step size control of the numerical solutions. Itwould become very important for avoiding undesirable explosive realizations inthose situations where the SDE (1) has unstable solutions.

Another aspect to take into account in the implementation of the continuous QRalgorithm is the evaluation of the integrals in (25). Firstly it should be noted thatthe expression (18) can be rewritten as

ψj (x,Q) = tr(Q⊤Df0 (x)Q

)|jj+

1

2

m∑

i=1

[tr(Q⊤D (Dfi (x) fi (x))Q

)|jj

+tr(Q⊤ (Dfi (x))⊤(Id −Q|j(Q|j)⊤)Dfi (x)Q)|jj ]

−(tr(Q⊤Dfi (x)Q

)|jj)2,

which is easier to implement numerically and allows the computation of all ∆ψj ,j = 1, .., d at once. Additionally, the trivial case ψd (x,Q) does not depend on Q

and, for linear SDEs, it is a constant function that only depends on the jacobianDfi, i = 0, ...,m evaluated at x = 0. In other words, the continuous QR methodproduces exact results while computing the sum of all LEs corresponding to linearSDEs. Here, we used the composite trapezoidal rule for approximating the integralstn+1∫tn

∆ψj(xu, Qu)du, n ≥ 0, j = 1, .., d − 1. However, an alternative approach can

be obtained by the numerical integration of the extended system

dxt =

m∑

i=0

fi (xt) dwit, xt0 = x0,

dQt =m∑

i=0

QtTit (xt,Qt) dwi

t, Qt0 = Id,

dzjt = ∆ψj (xt,Qt) dw0t , zt0 = ∆ψj (x0, Id) , j = 1, ..., d,


for which one has

λj(T ) =1

T − t0

nT−1∑

n=0

tn+1∫

tn

∆ψj(xu, Qu)du

=zjT

T − t0.

Finally, some considerations about the choice of the truncation time T are inorder. In [36] it was enunciated a Central-Limit theorem (Theorem 5.2) as a possiblesolution to this problem. However, as it was noted there, this approach is extremelydifficult to use in practice because the variance of the limit laws depends on theunknown LEs. Alternatively, it was proposed the heuristic criterion of taking T =

t0+Nh such that λj (t0 + nh) be almost constant for n aroundN . However, usually

one does not expect large changes in λj (t0 + nh) from n to n+1.A different criterionintroduced in [13] could be followed. The basic idea is to take T = T0 such that,

for given ∆T > 0 large enough, the approximations λj (T0 + k∆T ), k = 0, 1, ...,becomes almost constant with respect to k.

6. Numerical Examples

In this section, the performance of the QR methods is illustrated through twoexamples. For the Discrete and Continuous Projected QR-methods, the classicalEuler and Milstein weak integrators [25] for SDEs were used, which will be denotedby ”D − Euler”, ”D −Milstein”, ”CP − Euler”, and ”CP −Milstein”, respec-tively. These integrators were also used in combination with the 2 -stage automaticorthogonal Runge-Kutta scheme [14] to integrate, respectively, the equations (20)and (21) for the Continuous QR method. These Continuous Automatic methodswill be called ”CA− Euler −RK2” and ”CA−Milstein−RK2”.

Example 1. Let us consider the nonlinear 1-dimensional Ito SDE given by

(38) dxt = (−axt + F (xt)) dt+G (xt) dwt,

where F (x) = arctan (x) and G (x) =√1 + x2. For this equation the LE can be

explicitly computed by the expression

λ = −a+∫

R

(F ′ (x) − 1

2G′ (x)

2

)p (x) dx,

for any value of the parameter a, where p (x) is the invariant probability law of xt[19]. For example, for a = 2 the ”true ” LE is given by λ = −1.3385.

The Table I summarizes the values of the approximate LE λ(T ) computed bythe discrete and the continuous QR methods at T = 500 for different step sizes.For each step size we also included the relative CPU, which is obtained as the ratioof the actual CPU time for each numerical scheme to the minimum of all CPUtimes. Note that the discrete QR methods are the less computationally expensive,in contrast to the continuous automatic variants, which are up to 3.5 times slower.Since we have not proved almost sure convergence to the true LEs, we included

the values of E(λ(T )), estimated on the base of 100 independent realizations. As

expected, the values of E(λ(T )) are closer to the true λ than the approximation

obtained with a single realization of λ(T ). However, despite no almost sure con-

vergence was proven, we can easily observe that single realizations of λ(T ) alsoprovide a very accurate approximation to λ. Therefore, we conjecture that our


0 100 200 300 400 500−1.42

−1.4

−1.38

−1.36

−1.34

−1.32

−1.3

−1.28

−1.26

0 100 200 300 400 500−1.42

−1.4

−1.38

−1.36

−1.34

−1.32

−1.3

−1.28

−1.26

Time Time

Lyap

unov

Exp

onen

t

Lyap

unov

Exp

onen

t

TrueD−EulerPC−EulerAC−Euler−RK2

TrueD−MilsteinPC−MilsteinAC−Milstein−RK2

Figure 1. QR-based approximations to the LEs λ(T ) of the Ex-ample 1, implemented with a) the Euler integrator, and b) theMilstein integrator, with step size h = 2−9 and T = 500.

approximations λ(T ) are in fact LEs of the discrete numerical schemes, which con-verge almost surely to the true λ. However, proving such kind of results is going tobe subject of another work in the near future.

Method Step Size h λ(T ) E(λ(T )) Rel. CPU

D − Euler2−6

2−9−1.3638−1.3491

−1.3635−1.3433

11

D −Milstein2−6

2−9−1.3576−1.3409

−1.3556−1.3407

1.00421.0115

CP − Euler2−6

2−9−1.3492−1.3446

−1.3411−1.3389

1.32981.3392

CP −Milstein2−6

2−9−1.3455−1.3303

−1.3379−1.3395

1.33271.3409

CA− Euler −RK22−6

2−9−1.3281−1.3309

−1.3383−1.3359

1.68151.7201

CA−Milstein−RK22−6

2−9−1.3297−1.3404

−1.3390−1.3387

3.56553.5722

Table I. QR-based approximations to λ = −1.3385 of Example 1, computed for two

different step sizes with T = 500. For each method, relative CPU times (ratio of the

actual CPU time to the minimum of all CPU times) are also reported.

Figure 1 shows the value of the approximate LE λ(T ), 0 ≤ T ≤ 500, obtainedby the QR-methods with step-size h = 2−9. The left panel corresponds to themethods D−Euler, CP −Euler and CA−Euler−RK2. Here, the CP −Eulerand the CA−Euler−RK2 methods seems to be those of the better performance.Whereas, the right panel corresponds to the methods D−Milstein, CP −Milsteinand CA−Milstein−RK2, which also suggest that better results are obtained bythe continuous QR methods.


−1.45

−1.338

−1.25

λ

−1.45

−1.338

−1.25

λ

−1.45

−1.338

−1.25

λ

T=100 T=200 T=300 T=400 T=500

h=2−4

h=2−6

h=2−8

Figure 2. Boxplots corresponding to 100 trials of LE λ(T ) forthe Example 1. The trials were obtained by the methods D-Milstein, CP-Milstein and CA-Milstein-RK2 with step-size h =2−4, 2−6, 2−8 and truncation times T = 100, 200, 300, 400, 500.

This is better illustrated in Figure 2, which shows the mean value λ(T ) =

1100

100∑i=1

λi(T ) of the approximated LEs λi(T ), i = 1, .., 100, computed by the meth-

ods D −Milstein, CP −Milstein and CA −Milstein−RK2 in 100 trials, withstep-size h = 2−4, 2−6, 2−8 and truncation times T = 100, 200, 300, 400, 500. Eachsubplot in the figure shows a box and whisker plot (boxplot) with one box for each

of the methods. The boxes has lines at the mean value λ(T ) and at the 95% con-

fidence interval [λ(T ) − 1.96σλ(T ) , λ(T ) + 1.96σλ(T )], where σλ(T ) denotes the

standard deviation of the 100 values λi(T ). The whiskers extending from each end

of the boxes show the minimum and the maximum value of λi(T ), i = 1, ..., 100. Inthis example, for h relatively large (h = 2−4) the method D−Milstein presents theworse performance; even more, this situation does not improves with the increasingof T . Note also that for this step size, the continuous QR method CP −Milsteinachieves the best results. It can be also seen that for the other two step sizes, the

estimation of λ(T ) remains almost with the same value for the three methods. Onthe other hand, as expected, the length of the confidence interval (determined by

the standard deviation σλ(T )) decreases with the increasing of T . At glance, λ(T )seems to be a constant for all values of T . However, it gets closer to the true λwhen T increases, which is shown in the next figure.

Indeed, Figure 3 shows the errors eλ(T ) =∣∣∣λ− λ(T )

∣∣∣, T = 200, 300, ..., 600, for

the CP −Milstein method with step size h = 2−6, where λ(T ) was estimated from

500 values λi(T ). For these values of eλ(T ), a nonlinear regression model of theform

eλ(T ) = a+ b(1/T )c


2 2.5 3 3.5 4 4.5 5

x 10−3

0.6

0.8

1

1.2

1.4

1.6

1.8

2x 10

−3

1/T

Eλ(T

)

Figure 3. Plot of the the errors eλ(T ) =∣∣∣λ− λ(T )

∣∣∣, T =

200, 300, ..., 600, for the method PC − Milstein with step sizeh = 2−6 and the estimated curve eλ(T ) = 2.54 × 10−4 +0.43(1/T )1.04.

−7.5 −7 −6.5 −6 −5.5 −5 −4.5 −4 −3.5−13

−12.5

−12

−11.5

−11

−10.5

−10

−9.5

−9

−8.5

log2(h)

log 2(E

λ(h))

Figure 4. Plot of the errors log2(eλ(h)) = log2(|λ − λ(500)|)for the method PC − Milstein method with step sizes h =2−4, 2−5, 2−6, 2−7 and the estimated curve log2(eλ(h)) = −4.89 +1.03 log2(h).

was fitted. This yields the estimates a = 2.54× 10−4, b = 0.43 and c = 1.04. The

figure also shows the curve a+ b(1/T )c for T ∈ [200, 600]. Notice that the estimatedvalue of c agrees with the theoretical value c = 1 provided by Theorem 4.1.

On the other hand, the errors eλ(h) =∣∣∣λ− λ(500)

∣∣∣ were also computed for the

CP −Milstein method with step sizes h = 2−4, 2−5, 2−6, 2−7 and T = 500, where


λ(T ) was estimated from 500 values λi(T ). A regression model of the form

log2(eλ(h)) = α+ γ log2(h)

was fitted, which yields to the estimations α = −4.89 and γ = 1.03. Figure 4shows these error values and the curve α + γ log2(h). As expected, the estimatedvalue of γ is very close to the theoretical value γ = 1, which is the weak order ofconvergence of the weak Milstein scheme.

Example 2. The second example corresponds to the damped harmonic oscillatorthat is defined by the 2-dimensional linear SDE

(39) dxt =

(0 1−α 2β

)xtdt+

(0 0σ 0

)xt dwt,

where β is the damping constant and the parameter α controls the strength of the

restoring force. It is known from [22] and [23] that, for |β| = √α and γ = σ2

2 , theLyapunov exponents are given by

λ1 = β + 121/3Γ (1/2)

Γ (1/6)

γ1/3

2and λ2 = β − 121/3

Γ (1/2)

Γ (1/6)

γ1/3

2.

In this example, two different sets of values for β and σ were selected. The

first one is σ = 1 and β = −121/3 Γ(1/2)Γ(1/6)

γ1/3

2 , which yield the LEs λ1 = 0 and

λ2 = −0.5786. The second one corresponds to σ = 1 and β = 121/3 Γ(1/2)Γ(1/6)

γ1/3

2 ,

resulting in the LEs λ1 = 0.5786 and λ2 = 0. Note that, for this linear equation,the fixed point x0= 0 generates a stationary orbit for the cocycle ϕ and the Diracmeasure µ = δx0

is ϕ -invariant. Then, the linear matrix cocycle Dϕ can becomputed directly from (2) in such a way of avoiding the numerical integration ofthe equation (39) (i.e. setting xt ≡ 0 in (2)). Nevertheless, in order to illustrate theperformance of the proposed numerical methods on general SDEs with unknowninvariant measure, the equation (39) was solved numerically as well, where the pointx0 = (0, 0.1) was used as initial condition.

For the first set of parameters, Table II shows the performance of the QR meth-ods for two different step sizes.

Method Step Size h λ1(T ) λ2(T ) Σλ(T ) Rel. CPU

D-Euler2−6

2−90.05730.0103

−0.6374−0.6014

−0.5801−0.5911

11

D-Milstein2−6

2−9−0.01410.0104

−0.5523−0.5901

−0.5664−0.5797

1.01581.0030

CP-Euler2−6

2−90.02370.0132

−0.6023−0.5918

−0.5786−0.5786

1.34381.1977

CP-Milstein2−6

2−9−0.0079−0.0018

−0.5707−0.5767

−0.5786−0.5786

1.36011.2007

CA-Euler-RK22−6

2−9−0.0181−0.0012

−0.5604−0.5773

−0.5786−0.5786

3.20362.1503

CA-Milstein-RK22−6

2−9−0.0109−0.0011

−0.5676−0.5775

−0.5786−0.5786

3.22232.1540

Table II. QR-based approximations to the LEs λ1 = 0and λ2 = −0.5786 of Example 2,

computed for two different step sizes with T = 500. For each method, relative CPU times


0 100 200 300 400 500−0.8

−0.6

−0.4

−0.2

0

0.2

0 100 200 300 400 500−0.8

−0.6

−0.4

−0.2

0

0.2

TimeTimeLy

apun

ov E

xpon

ent

Lyap

unov

Exp

onen

t


TrueD−EulerCP−EulerCA−Euler−RK2

Figure 5. QR-based approximations to the LEs λ1(T ) and λ2(T )of the Example 2, implemented with a) the Euler integrator, andb) the Milstein integrator, with step size h = 2−9, T = 500 and

parameters σ = 1, β = −61/3 Γ(1/2)2Γ(1/6) .

(ratio of the actual CPU time to the minimum of all CPU times) are also reported.

Similarly to Figure 1, Figure 5 shows the approximate LEs λ1(T ), λ2(T ), 0 ≤T ≤ 500, for different numerical schemes. As in the previous example above, thecontinuous methods seem to be those with best performance.

For the second set of parameters,Table III shows the results obtained by the QRmethods with two different step sizes.

Method Step Size h λ1(T ) λ2(T ) Σλ(T )

D-Euler2−6

2−90.55820.5898

0.0203−0.0130

0.57850.5768

D-Milstein2−6

2−90.58690.5844

−0.0094−0.0064

0.57750.5780

CP-Euler2−6

2−90.59220.5603

−0.01360.0183

0.57860.5786

CP-Milstein2−6

2−90.59110.5895

−0.0125−0.0109

0.57860.5786

CA-Euler-RK22−6

2−90.56280.5695

0.01580.0091

0.57860.5786

CA-Milstein-RK22−6

2−90.58940.5823

−0.0108−0.0037

0.57860.5786

Table III. QR-based approximations to the LEs λ1 = 0.5786and λ2 = 0 of Example 2,

computed for two different step sizes with T = 500.


0 100 200 300 400 500−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 100 200 300 400 500−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

TimeTimeLy

apun

ov E

xpon

ent

Lyap

unov

Exp

onen

t


TrueD−EulerPC−EulerAC−Euler−RK2

Figure 6. QR-based approximations to the LEs λ1(T ) and λ2(T )for the Example 2, implemented with a) the Euler integrator, andb) the Milstein integrator, with step size h = 2−9, T = 500 and

parameters σ = 1, β = 61/3 Γ(1/2)2Γ(1/6) .

Figure 6 shows the approximate LEs λ1(T ), λ2(T ), 0 ≤ T ≤ 500, in this case.Note that it is very similar to Figures 1 and 5. None of the numerical realiza-tions corresponding to h = 2−6, 2−9 produced explosive behavior in any numericalintegrator.

It is easy to check from Tables II and III that all continuous QR methods auto-matically preserve the Lyapunov regularity condition. On the other hand, althoughnot exacts, discrete QR methods provide good approximations to the sum of all LEs.

Example 3. The final example corresponds to the stochastic Lorenz system [24],which is given by the SDE

dxt = (f +Bxt + F(xt)) dt+ σxt dwt,

where

f =

00

−b(r + s)

, B =

−s s 0−s −1 00 0 −b

, F(x) =

0−x1x3

x1x2

,

s, r, b, σ are positive constants and f is an external force. As in [24], we concentrateon the parameter values s = 10, b = 8

3 and study changes on the Lyapunov spectrumfor different values of r > 0.

As it was shown in [24], two different dynamical behaviors can be obtained byvarying r. Namely, for small values of r all LEs are negative, which correspondsto a one point random attractor. While increasing r, the top LE reaches zero(indicating a stochastic bifurcation) to finally become positive for large values of r.

It is worth mentioning that, despite being defined by a nonlinear SDE, the Lorenzsystem is a special case for which ψd (x,Q) is a constant function. Indeed,

ψd (x,Q) = tr(B+DF(x)) +1

2(tr(σ2I3)− tr(σI3)

2

= −s− 1− b− 3σ2 = −13.9367.


Thus, it is another example for which the continuous QR method is exact whileapproximating the sum of all LEs.

Table IV shows the computed LEs (T = 500) for two extreme regimes (r = 5and r = 25) corresponding to σ = 0.3 and step size h = 2−8.

Method λ(T ), r = 5 λ(T ), r = 25

D-Euler −0.688 −1.073 −12.172 0.644 −0.106 −14.239

D-Milstein −0.618 −1.151 −12.200 0.637 −0.096 −14.304

CP-Euler −0.689 −1.186 −12.062 0.487 −0.364 −14.060

CP-Milstein −0.648 −1.209 −12.079 0.425 −0.366 −13.996

CA-Euler-RK2 −0.659 −1.194 −12.084 0.450 −0.371 −14.016

CA-Milstein-RK2 −0.708 −1.145 −12.083 0.442 −0.372 −14.007

Table IV. QR-based approximations to the LEs of Example 3 for two different values

of r,with h = 2−8 and T = 500.

As seen on the table, all QR-based methods reproduce well the two different dy-namical behaviors corresponding to r = 5 and r = 25, and the Lyapunov regularityproperty.

7. Conclusions

In this work, two numerical methods for approximating the Lyapunov Exponentsof stochastic differential equations were introduced. To the authors knowledge, thiswork is the first attempt for a systematic study of the QR-based methods in thestochastic case. Such methods constitutes a stochastic version of the well-knownQR−based methods that have been long used for ordinary differential equations.In contrast with previous works reported in the literature, the QR−based methodspresented here perform well for the approximation of the all Lyapunov exponents.

The numerical examples show that the continuous QR methods seem to be theones with best performance. This fact can be also corroborated from the errors anal-ysis carried out in section 4. There, it was shown that some difficulties may occurin the computation of negative LEs of large magnitude with discrete QR methods.This limitation can be avoided through the use of continuous QR methods, but atthe expense of increasing the computational complexity.

Finally, it is worth to remark that although the present work follows essentiallythe same ideas exposed in [17], it can be also extended for adapting some promisingresults about continuous QR methods in ODEs [12] to the stochastic case. Namely,the computation of just a few largest LEs of an SDE by means of the numericalintegration of a weak skew-symmetric stochastic SDE. This issue and some otherextensions of the theory of the QR-based methods for SDEs will be subject of afuture work.

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Postdoctoral fellow, Montreal Neurological Institute, McGill University, Montreal, CanadaE-mail : [email protected], [email protected]

CIMFAV-DEUV, Faculty of Sciences, Valparaiso University, ChileE-mail : [email protected]

ICIMAF, Interdisciplinary Mathematics Department, Havana, CubaE-mail : [email protected]

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