Almost sure exponential stability in the numerical simulation of stochastic differential equations
Mao, Xuerong (2015) Almost sure exponential stability in the numerical simulation of stochastic differential equations. SIAM Journal on Numerical Analysis, 53 (1). pp. 370-389. ISSN 0036-1429 (https://doi.org/10.1137/140966198)
Preview |
Text.
Filename: Mao_SIAM_JNA_2015_Almost_sure_exponential_stability_in_the_numerical_simulation.pdf
Final Published Version License: Download (468kB)| Preview |
Abstract
This paper is mainly concerned with whether the almost sure exponential stability of stochastic differential equations (SDEs) is shared with that of a numerical method. Under the global Lipschtiz condition, we first show that the SDE is pth moment exponentially stable (for p 2 (0; 1)) if and only if the stochastic theta method is pth moment exponentially stable for a sufficiently small step size. We then show that the pth moment exponential stability of the SDE or the stochastic theta method implies the almost sure exponential stability of the SDE or the stochastic theta method, respectively. Hence, our new theory enables us to study the almost sure exponential stability of the SDEs using the stochastic theta method, instead of the method of the Lyapunov functions. That is, we can now carry out careful numerical simulations using the stochastic theta method with a sufficiently small step size t. If the stochastic theta method is pth moment exponentially stable for a sufficiently small p 2 (0; 1), we can then infer that the underlying SDE is almost surely exponentially stable. Our new theory also enables us to show the ability of the stochastic theta method to reproduce the almost sure exponential stability of the SDEs. In particular, we give positive answers to two open problems (P1) and (P2) listed on page 2.
ORCID iDs
Mao, Xuerong ORCID: https://orcid.org/0000-0002-6768-9864;-
-
Item type: Article ID code: 50887 Dates: DateEvent3 February 2015Published3 December 2014AcceptedSubjects: Science > Mathematics > Probabilities. Mathematical statistics Department: Faculty of Science > Mathematics and Statistics Depositing user: Pure Administrator Date deposited: 05 Jan 2015 11:03 Last modified: 05 Sep 2024 00:48 Related URLs: URI: https://strathprints.strath.ac.uk/id/eprint/50887