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arXiv:math/0502317v1 [math.PR] 15 Feb 2005

An adaptive scheme for the approximation of

dissipative systems

Vincent Lemaire

Laboratoire d’Analyse et de Math´ ematiques Appliqu´ ees, UMR 8050,

Universit´ e de Marne-la-Vall´ ee, 5 boulevard Descartes, Champs-sur-Marne,

F-77454 Marne-la-Vall´ ee Cedex 2, France.

Abstract

We propose a new scheme for the long time approximation of a diffusion when the

drift vector field is not globally Lipschitz. Under this assumption, regular explicit

Euler scheme –with constant or decreasing step– may explode and implicit Euler

scheme are CPU-time expensive. The algorithm we introduce is explicit and we

prove that any weak limit of the weighted empirical measures of this scheme is

a stationary distribution of the stochastic differential equation. Several examples

are presented including gradient dissipative systems and Hamiltonian dissipative

systems.

Key words: diffusion process; dissipative system; invariant measure; stochastic

algorithm; Euler method; simulation

1991 MSC: 65C30, 60J60

1Introduction

We consider the following stochastic differential equation

dxt= b(xt)dt + σ(xt)dBt,x(0) = x0∈ Rd, (1)

where b : Rd→ Rdis a locally Lipschitz continuous vector field and σ is locally

Lipschitz continuous on Rd, with values in the set of d × m matrices and B

is an m-dimensional Brownian motion. Assume that (xt)t≥0has a Lyapounov

function V i.e. a positive regular function decreasing along trajectories (precise

conditions are given by Assumption 1 in Section 2), so that there exists at least

one invariant measure.

Email address: vincent.lemaire@univ-mlv.fr (Vincent Lemaire).

Preprint submitted to Elsevier Science1 February 2008

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Until recently, the approximation of the stationary mode of the diffusion has

been studied under the assumption that V is essentially quadratic i.e.

|∇V |2= O(V )andsup

x∈Rd?D2V ? < +∞, (2)

and |b|2= O(V ) (which implies sublinear growth for b). When σ is bounded

and the diffusion is uniformly strictly elliptic, the invariant measure ν is unique

and Talay proposed in [12] a method for the computation of ν based on the

constant step Euler scheme. He proved the convergence of the invariant mea-

sure of the scheme to ν. On the other hand, Lamberton and Pag` es studied in

[5] the ergodic properties of the weighted empirical measures (νη

creasing step Euler scheme. They proved the almost sure tightness of (νη

and that any weak limit is a stationary distribution for the diffusion.

n)n≥0of a de-

n)n≥1

However, the conditions (2) and |b|2= O(V ) are too restrictive for studying

systems used in random mechanics (see Soize [10]). Indeed, the drift vector

field b is generally locally Lipschitz and in many cases V is not essentially

quadratic. This framework has been recently investigated by Talay in [13]

and by Mattingly et al. in [9]. In these papers, implicit Euler schemes with

constant steps are used for the approximation of the diffusion. In recent work,

Lamba, Mattingly and Stuart have introduced on finite time interval [0;T] an

adaptive explicit Euler scheme (see [4] and [8]). The step is adapted according

to the error between the Euler and Heun approximations of the ODE ˙ x = b(x).

They prove strong mean-quadratic convergence of the scheme on over finite

time intervals and ergodicity when the noise is non-degenerate. We propose a

completely different explicit scheme based on a stochastic step sequence and

we obtain the almost sure convergence of its weighted empirical measures to

the invariant measure of (1).

The key to prove the almost sure tightness of the weighted empirical measures

of the decreasing Euler scheme (Xn)n≥0introduced in [5] is that the scheme

satisfies a stability condition i.e. there exist ˜ α > 0 and˜β > 0 such that

E

?

V (Xn+1)|Fn

?

− V (Xn)

γn+1

≤ −˜ αV (Xn) +˜β,(3)

where (γn)n≥0is the deterministic decreasing step sequence. Without assump-

tions (2) we can no longer prove the stability condition (3) for this scheme.

Our scheme is built in order to satisfy (3). We proceed as follows. Firstly, we

start from a deterministic X0= x0∈ Rdand set

Xn+1= Xn+ ˜ γn+1b(Xn) +

?

˜ γn+1σ(Xn)Un+1,n ≥ 0, (4)

where (Un)n≥1 is a Rm-white noise more precisely defined in Section 2 and

˜ γn+1 = γn+1∧ χn with (γn)n≥0a positive nonincreasing sequence and χn a

σ(U1,...,Un)–measurable random variable.

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The basic main idea is to choose χnsmall when the scheme starts to explode.

In this case the discretization is finer and the stability condition of the diffusion

prevents the explosion. Furthermore we prove that the scheme satisfies a sim-

ilar condition with (3). A non-optimal –although natural– choice for χnmay

be χn=

the Lyapounov function (see (29) and (53)).

1

|b(Xn−1)|2∨1. For the two studied examples, optimal choices depend on

A crucial feature of our algorithm is the existence of an almost surely finite

time n1such that for every n ≥ n1, ˜ γn= γni.e. the event {χn−1< γn} does

not occur any more.

Numerically the algorithm is very simple to implement and the complexity is

the same as that of a regular Euler scheme. Another interest is that the scheme

is explicit, which is a big advantage on implicit schemes for high dimensional

problems. Indeed a fixed point algorithm is not needed is our case. Moreover,

we will see that wrong convergence problem due to fixed point algorithm may

be avoided using our algorithm.

The paper is organized as follows. We introduce the framework and the algo-

rithm in Section 2. In Section 3 are presented some preliminary results about

the approximation scheme of (Xn)n≥0defined in (4). In Section 4 we extend

some results of [5] and give conditions for the almost sure tightness of the

empirical measure and for its weak convergence to an invariant measure of

(1). Section 5 is devoted to the study of monotone systems and Section 6 of

stochastic Hamiltonian dissipative systems. The numerical experiments are

in Section 7 including some comparaison with recently introduced implicit

scheme. We confirm the non-explosion and the convergence of the scheme.

2 Framework and algorithm

We will denote by A the infinitesimal generator of (1). The following assump-

tion will be needed throughout the paper.

Assumption 1 There is a C2function V on Rdwith values in [1,+∞[ such

that lim

|x|→+∞V (x) = +∞ and satisfying

∃α > 0, ∃β > 0, ?∇V,b? ≤ −αV + β, (5)

∃CV,σ> 0, ∃a ∈ (0,1],

and∃C > 0,

Tr

?

?

σ∗(∇V )⊗2σ

σ∗D2V σ

?

≤ CV,σV2−a,

(x) ≤ C sup sup

x∈RdTr

?

x∈RdTr(σ∗σ)(x).

(6)

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Remark 1 If V is essentially quadratic (i.e. satisfies (2)) then (6) is satisfied

as soon as there exists Cσ> 0 and a ∈ (0,1] such that Tr(σ∗σ) ≤ CσV1−a.

Under this assumption, there exists a global solution to equation (1) and

(at least) one invariant measure. An important point to note here is that all

invariant measures have exponential moments. Indeed, an easy computation

shows that for all λ <

α

aCV,σwe have

∃˜ α > 0, ∃˜β > 0,Aexp

?

λVa?

≤ −˜ αexp(λVa) +˜β,

and this implies ν

?

exp(λVa)

?

is finite for all invariant measures ν.

For the approximation of the diffusion, we assume that (Un)n≥1is a sequence

of i.i.d. random variables defined on a probability space (Ω,A,P), with values

in Rm, and such that U1is a generalized Gaussian (see Stout [11]) i.e.

∃κ > 0, ∀θ ∈ Rd,

E

?

exp

?

?θ,U1?

??

≤ exp

?κ|θ|2

2

?

. (7)

and that var(U1) = Idm. We will call (Un)n≥1a Rm-valued generalized Gaus-

sian white noise. The condition (7) implies that U1is centered and satisfies

∃τ > 0,

E

?

exp

?

τ|U1|2??

< +∞. (8)

Moreover, the condition var(U1) = Idm implies that κ ≥ 1. In the sequel,

Fndenotes, for n ≥ 1, the σ-field generated on Ω by the random variables

U1,...,Un, and F0the trivial σ-field.

Remark 2 The assumptions made on the white noise (Un)n≥1are not restric-

tive for numerical implementation. Indeed, centered Gaussian and centered

bounded random variables satisfy (7).

The stochastic step sequence ˜ γ = (˜ γn)n≥0is defined by

∀n ≥ 1,˜ γn= γn∧ χn−1,˜ γ0= γ0,(9)

where (γn)n≥0is a deterministic nonincreasing sequence of positive numbers

satisfying

limand

nγn= 0

?

n≥0

γn= +∞,

and (χn)n≥1is an (Fn)n≥0–adapted sequence of positive random variables. It

is important to note that the step sequence ˜ γ is (Fn)n≥0–predictable.

Now we introduce the weighted empirical measures like Lamberton and Pag` es

in [5]. Given a sequence η = (ηn)n≥1of positive numbers satisfying?

n≥1ηn=

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+∞, we denote by νη

nthe random probability measure on Rddefined by

νη

n=

1

Hn

n

?

k=1

ηkδXk−1, withHn=

n

?

k=1

ηk.

Throughout the paper, |.| denotes the Euclidean norm and ?.? denotes the

natural matrix norm induced by |.| i.e. for every square matrix A, ?A? =

sup|x|=1|Ax|. The letter C is used to denote a positive constant, which may

vary from line to line.

3 Preliminary results

In this section, we prove results which are the keys to study the Euler scheme

with predictable random step defined in the introduction. Proposition 3 con-

tains two results: the first one (11) provides a substitute for the Lp–bounded-

ness of (V (Xn))n≥0used in [5]. The second one (12) is a new consequence of

the stability condition (10) and is used to prove the fundamental proposition

5 which ensures the existence of an almost surely finite time n1such that for

every n ≥ n1, ˜ γn= γn.

Proposition 3 Let W be a nonnegative function and (˜ γn)n≥0be a (Fn)n≥0–

predictable sequence of positive and finite random variables satisfying: there

exist α > 0, β > 0, n0∈ N, such that

?

˜ γn+1

∀n ≥ n0,

E

W(Xn+1)|Fn

?

− W(Xn)

≤ −αW(Xn) + β. (10)

Suppose (θn)n≥1is a positive nonincreasing sequence such that E

is finite, then

?

n≥n0+1

If, in addition, limnθn= 0 then

??

n≥0θn˜ γn

?

E

?

θn˜ γnW(Xn−1)|Fn0

?

< +∞.(11)

lim

nθnW(Xn) = 0 a.s.(12)

The above proposition is related to Robbins-Siegmund’s theorem (see Theorem

1.3.12 in [1] and the references therein).

Proof. Let Rn=

n

?

k=0

θk˜ γkand R∞= lim

nRn∈ L1(P).

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