Numerical method of highly nonlinear and nonautonomous neutral stochastic differential delay equations with Markovian switching

In this paper, we establish a partially truncated Euler–Maruyama scheme for highly nonlinear and nonautonomous neutral stochastic differential delay equations with Markovian switching. We investigate the strong convergence rate and almost sure exponential stability of the numerical solutions under the generalized Khasminskii-type condition.

Optimal rate of convergence for two classes of schemes to stochastic differential equations driven by fractional Brownian motions

This paper investigates numerical schemes for stochastic differential equations driven by multi-dimensional fractional Brownian motions (fBms) with Hurst parameter $H\in (\frac 12,1)$. Based on the continuous dependence of numerical solutions on the driving noises, we propose the order conditions of Runge–Kutta methods for the strong convergence rate $2H-\frac 12$, which is the optimal strong convergence rate for approximating the Lévy area of fBms. We provide an alternative way to analyse the convergence rate of explicit schemes by adding ‘stage values’ such that the schemes are interpreted as Runge–Kutta methods. Taking advantage of this technique the strong convergence rate of simplified step-$N$ Euler schemes is obtained, which gives an answer to a conjecture in Deya et al. (2012) when $H\in (\frac 12,1)$. Numerical experiments verify the theoretical convergence rate.

Strong approximation of monotone stochastic partial differential equations driven by white noise

We establish an optimal strong convergence rate of a fully discrete numerical scheme for second-order parabolic stochastic partial differential equations with monotone drifts, including the stochastic Allen–Cahn equation, driven by an additive space-time white noise. Our first step is to transform the original stochastic equation into an equivalent random equation whose solution possesses more regularity than the original one. Then we use the backward Euler in time and spectral Galerkin in space to fully discretise this random equation. By the monotonicity assumption, in combination with the factorisation method and stochastic calculus in martingale-type 2 Banach spaces, we derive a uniform maximum norm estimation and a Hölder-type regularity for both stochastic and random equations. Finally, the strong convergence rate of the proposed fully discrete scheme is obtained. Several numerical experiments are carried out to verify the theoretical result.

Analytical and numerical aspect of coincidence point problem of quasi-contractive operators

We propose a new Jungck-S iteration method for a class of quasi-contractive operators on a convex metric space and study its strong convergence, rate of convergence and stability. We also provide conditions under which convergence of this method is equivalent to Jungck-Ishikawa iteration method. Some numerical examples are provided to validate the theoretical findings obtained herein. Our results are refinement and extension of the corresponding ones existing in the current literature.

Exponential inequalities under sub-linear expectations with applications to strong law of large numbers

In this paper, we give some exponential inequalities for extended independent random variables under sub-linear expectations. As an application, we obtain the strong convergence rate O(n-1/2 ln1/2 n) for the strong law of large numbers under sub-linear expectations, which generalizes some corresponding ones under the classical linear expectations.

Strong convergence rates of semidiscrete splitting approximations for the stochastic Allen–Cahn equation

This article analyses an explicit temporal splitting numerical scheme for the stochastic Allen–Cahn equation driven by additive noise in a bounded spatial domain with smooth boundary in dimension $d\leqslant 3$. The splitting strategy is combined with an exponential Euler scheme of an auxiliary problem. When $d=1$ and the driving noise is a space–time white noise we first show some a priori estimates of this splitting scheme. Using the monotonicity of the drift nonlinearity we then prove that under very mild assumptions on the initial data this scheme achieves the optimal strong convergence rate $\mathcal{O}(\delta t^{\frac 14})$. When $d\leqslant 3$ and the driving noise possesses some regularity in space we study exponential integrability properties of the exact and numerical solutions. Finally, in dimension $d=1$, these properties are used to prove that the splitting scheme has a strong convergence rate $\mathcal{O}(\delta t)$.