Almost sure stability of the Euler–Maruyama method with random variable stepsize for stochastic differential equations

Abstract

In this paper, the Euler-Maruyama (EM) method with random variable stepsize is studied to reproduce the almost sure stability of the true solutions of stochastic differential equations.Since the choice of the time step is based on the current state of the solution, the time variable is proved to be a stopping time. Then the semimartingale convergence theory is employed to obtain the almost sure stability of the random variable stepsize EM solution. To our best knowledge, this is the first paper to apply the random variable stepsize (with clear proof of the stopping time) to the analysis of the almost sure stability of the EM method.

Adaptive timestepping for pathwise stability and positivity of strongly discretised nonlinear stochastic differential equations

“…Since the innovation process directly samples trajectories of W , it is now necessary to ensure that each t n is an F t -stopping time, where (F) t≥0 is the natural filtration of W , in order that the appropriate semi-martingale convergence theory may be applied. The importance of this issue was raised for the first time in the context of Euler-Maruyama methods with random variable stepsizes in Liu & Mao [13].…”

“…As in [13], we note that the modified form of the sequence h n in (14) is motivated by the need for each timestep to be rational, which is automatically the case when the method is implemented on a finite state machine. Remark 1.…”

“…Lemma 6.7. Let {X n } n∈N be a solution of (13). For each n ∈ N there exists a nonrandom B n ∈ (0, ∞) such that E|X n+1 | ≤ B n , n ∈ N.…”

Strong convergence and asymptotic stability of explicit numerical schemes for nonlinear stochastic differential equations

“…For instance, Guo et al [7] showed that the partially truncated EM method can preserve the mean square exponential stability of the underlying SDEs. Liu and Mao [26] made use of the EM method with random variable stepsize to reproduce the almost sure stability of the underlying SDEs. Szpruch and Zhang [39] established the asymptotic stability properties for the tamed EM scheme and the projected scheme, making use of some Lyapunov functions.…”

On Cubic Difference Equations with Variable Coefficients and Fading Stochastic Perturbations

“…Several recent publications are devoted to the use of adaptive timestepping in a explicit Euler-Maruyama discretization of nonlinear equations: for example [3,9,12,11]. In [9] (see also [7]) it was shown that suitably designed adaptive timestepping strategies could be used to ensure strong convergence of order 1/2 for a class of equations with non-globally Lipschitz drift, and globally Lipschitz diffusion.…”