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

Discrete Razumikhin-type technique and stability of the Euler-Maruyama method to stochastic functional differential equations

A discrete stochastic Razumikhin-type theorem is established to investigate whether the Euler--Maruyama (EM) scheme can reproduce the moment exponential stability of exact solutions of stochastic functional differential equations (SFDEs). In addition, the Chebyshev inequality and the Borel-Cantelli lemma are applied to show the almost sure stability of the EM approximate solutions of SFDEs. To show our idea clearly, these results are used to discuss stability of numerical solutions of two classes of special SFDEs, including stochastic delay differential equations (SDDEs) with variable delay and stochastically perturbed equations

Mean-square convergence and stability of the backward Euler method for stochastic differential delay equations with highly nonlinear growing coefficients

Over the last few decades, the numerical methods for stochastic differential delay equations (SDDEs) have been investigated and developed by many scholars. Nevertheless, there is still little work to be completed. By virtue of the novel technique, this paper focuses on the mean-square convergence and stability of the backward Euler method (BEM) for SDDEs whose drift and diffusion coefficients can both grow polynomially. The upper mean-square error bounds of BEM are obtained. Then the convergence rate, which is one-half, is revealed without using the moment boundedness of numerical solutions. Furthermore, under fairly general conditions, the novel technique is applied to prove that the BEM can inherit the exponential mean-square stability with a simple proof. At last, two numerical experiments are implemented to illustrate the reliability of the theories

Order-one strong convergence of numerical methods for SDEs without globally monotone coefficients

To obtain strong convergence rates of numerical schemes, an overwhelming majority of existing works impose a global monotonicity condition on coefficients of SDEs. On the contrary, a majority of SDEs from applications do not have globally monotone coefficients. As a recent breakthrough, the authors of [Hutzenthaler, Jentzen, Ann. Probab., 2020] originally presented a perturbation theory for stochastic differential equations (SDEs), which is crucial to recovering strong convergence rates of numerical schemes in a non-globally monotone setting. However, only a convergence rate of order $1/2$ was obtained there for time-stepping schemes such as a stopped increment-tamed Euler-Maruyama (SITEM) method. As an open problem, a natural question was raised by the aforementioned work as to whether higher convergence rate than $1/2$ can be obtained when higher order schemes are used. The present work attempts to solve the tough problem. To this end, we develop some new perturbation estimates that are able to reveal the order-one strong convergence of numerical methods. As the first application of the newly developed estimates, we identify the expected order-one pathwise uniformly strong convergence of the SITEM method for additive noise driven SDEs and multiplicative noise driven second order SDEs with non-globally monotone coefficients. As the other application, we propose and analyze a positivity preserving explicit Milstein-type method for Lotka-Volterra competition model driven by multi-dimensional noise, with a pathwise uniformly strong convergence rate of order one recovered under mild assumptions. These obtained results are completely new and significantly improve the existing theory. Numerical experiments are also provided to confirm the theoretical findings.Comment: 33 pages, 2 figure