Consistency and asymptotic normality of stochastic Euler schemes for ordinary differential equations

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Abstract

General stochastic Euler schemes for ordinary differential equations are studied. We give proofs on the consistency, the rate of convergence and the asymptotic normality of these procedures.

Introduction

We study the consistency and asymptotic normality of stochastic Euler schemes which are designed to approximate ordinary differential equations. Euler schemes are often used to simulate stochastic differential equations. Fierro and Torres (2001) study the consistency of these schemes in the context of Itô stochastic differential equations. However, this idea can be used to approximate ordinary differential equations, too: Fierro and Torres (2007) consider a special kind of Euler approximation for a given ODE. In this paper, we generalize the idea: Let there be given the ODE system ẋ=F(t,x), x(0)=x0Rd, t[0,T], 0<T<. Then we approximate the solution x on a partition πN of [0,T] with a stochastic Euler scheme that is based on random variables F̃kN instead on F. This approach can be useful in applications where one aims at approximating the trajectory of such a solution x for a function F which is costly to evaluate, for instance, in the case where F is the sum of (finitely) many single functions fi, iI, i.e.  F=iIfi. The paper is organized as follows: In Section  2 we introduce the basic notions and regularity conditions of the model. In Section  3 we give consistency results for our general Euler scheme. We state results on the asymptotic normality of the procedure in Section  4. Appendix A contains two technical statements which we use in this article, the proof of one of these statements is deferred to the supplement (Krebs, 2015).

Section snippets

Preliminaries

We denote for p1 by p the p-norms on the d-dimensional Euclidean space and we abbreviate the indicator function (on a measurable set) by 1. Let TR+ be a finite time horizon and let F=(F1,,Fd):[0,T]×RdRd be a continuous vector valued function. F fulfills the following growth conditions w.r.t. the first and second coordinate for s,t[0,T] and for x,yRdF(s,x)F(t,x)1K1(1+x1)|st|,F(t,x)F(t,y)1K2xy1, where 0<K1,K2< are some positive constants. Let there be given the ODE ẋ=F(t,

Consistency and rate of convergence

We come to the first main result of this paper, this is the convergence in mean of the processes xˆN, NN+, namely

Theorem 1

L2-Convergence of xˆN to x

Let the sequence of stochastic processes (xˆN:NN+) be defined in Eqs.   (3), (4). Let x be the unique global solution to the ordinary differential equation. Then, there exists a constant 0<B< such thatsupt[0,T]xˆN(t)x(t)1L2(P)BΔN.

Proof

Throughout the proof we shall write for the Euclidean 1-norm on Rd. Furthermore, we set xkNxN(tkN) and xˆkNxˆN(tkN), for k=0,,KN. First,

Asymptotic normality of stochastic approximation procedures

In this section we prove the asymptotic normality of the stochastic Euler schemes for ODE approximations.

Theorem 3

Let {πN:NN+} be the sequence of dyadic partitions of [0,T], i.e. πN={Tk/2N:k=0,1,,2N}. Let F=(F1,,Fd) fulfill the regularity conditions from   (1), (2). Additionally, let each component of F be continuously differentiable w.r.t. the space coordinate, i.e. (t,x)xFi(t,x) is continuous for i=1,,d.

Furthermore, let the stochastic approximations F̃kN be regular in that for all NN+ and k=1,

Acknowledgment

The author gratefully acknowledges the financial support of the Fraunhofer ITWM which is part of the Fraunhofer-Gesellschaft zur Förderung der angewandten Forschung e.V.

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