Mean Square Polynomial Stability of Numerical Solutions to a Class of Stochastic Differential Equations

The exponential stability of numerical methods to stochastic differential equations (SDEs) has been widely studied. In contrast, there are relatively few works on polynomial stability of numerical methods. In this letter, we address the question of reproducing the polynomial decay of a class of SDEs using the Euler--Maruyama method and the backward Euler--Maruyama method. The key technical contribution is based on various estimates involving the gamma function

Strong convergence rates for backward Euler–Maruyama method for non-linear dissipative-type stochastic differential equations with super-linear diffusion coefficients

In this work, we generalize the current theory of strong convergence rates for the backward Euler–Maruyama scheme for highly non-linear stochastic differential equations, which appear in both mathematical finance and bio-mathematics. More precisely, we show that under a dissipative condition on the drift coefficient and superlinear growth condition on the diffusion coefficient the BEM scheme converges with strong order of a half. This type of convergence gives theoretical foundations for efficient variance reduction techniques for Monte Carlo simulations. We support our theoretical results with relevant examples, such as stochastic population models and stochastic volatility models

Strong convergence and stability of implicit numerical methods for stochastic differential equations with non-globally Lipschitz continuous coefficients

We are interested in the strong convergence and almost sure stability of Euler-Maruyama (EM) type approximations to the solutions of stochastic differential equations (SDEs) with non-linear and non-Lipschitzian coefficients. Motivation comes from finance and biology where many widely applied models do not satisfy the standard assumptions required for the strong convergence. In addition we examine the globally almost surely asymptotic stability in this non-linear setting for EM type schemes. In particular, we present a stochastic counterpart of the discrete LaSalle principle from which we deduce stability properties for numerical methods

Almost sure stability with general decay rate of neutral stochastic pantograph equations with Markovian switching

This paper focuses on the general decay stability of nonlinear neutral stochastic pantograph equations with Markovian switching (NSPEwMSs). Under the local Lipschitz condition and non-linear growth condition, the existence and almost sure stability with general decay of the solution for NSPEwMSs are investigated. By means of M-matrix theory, some sufficient conditions on the general decay stability are also established for NSPEwMSs