Almost sure stability of stochastic theta methods with random variable stepsize for stochastic differential equations

Abstract

In this paper, we use the non-negative discrete semimartingale convergence theorems to study the stochastic theta methods with random stepsizes to reproduce the almost sure stability of the exact solution of stochastic differential equations. Moreover, the choice of the stepsize in each step is based on the stochastic theta methods of random variable stepsize. In numerical experiments, we propose an algorithm that successfully use θ-Maruyama and θ-Milstein methods to simulate the numerical solutions of stochastic differential equations, reproduce the almost sure stability of exact solutions of SDEs and simulate the random variable stepsize in each timestep, and compared with constant stepsizes, random stepsize can speed up the decay process and reduce the iterations greatly.

Keywords:

• Stochastic theta methods
• almost sure stability
• semimartingale convergence theory
• variable stepsize
• numerical simulation

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was supported by the National Natural Science Foundation of China [grant number 11901058] and the Natural Science Foundation of Hubei Province [grant number 2021CFB543].