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

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 theta-Maruyama and theta-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.