Consistency and asymptotic normality of stochastic Euler schemes for ordinary differential equations
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 , , , . Then we approximate the solution on a partition of with a stochastic Euler scheme that is based on random variables instead on . This approach can be useful in applications where one aims at approximating the trajectory of such a solution for a function which is costly to evaluate, for instance, in the case where is the sum of (finitely) many single functions , , i.e. . 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 by the -norms on the -dimensional Euclidean space and we abbreviate the indicator function (on a measurable set) by . Let be a finite time horizon and let be a continuous vector valued function. fulfills the following growth conditions w.r.t. the first and second coordinate for and for where are some positive constants. Let there be given the ODE
Consistency and rate of convergence
We come to the first main result of this paper, this is the convergence in mean of the processes , , namely Theorem 1 Let the sequence of stochastic processes be defined in Eqs. (3), (4). Let be the unique global solution to the ordinary differential equation. Then, there exists a constant such that-Convergence of to
Proof Throughout the proof we shall write for the Euclidean 1-norm on . Furthermore, we set and , for . 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 be the sequence of dyadic partitions of , i.e. . Let fulfill the regularity conditions from (1), (2). Additionally, let each component of be continuously differentiable w.r.t. the space coordinate, i.e. is continuous for . Furthermore, let the stochastic approximations be regular in that for all and
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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