arXiv subjects

Xiaoyue Li

Publications and source records attributed to Xiaoyue Li.

39 records · Page 3Linked to original sources

arXiv2020

The Strong Convergence and Stability of Explicit Approximations for Nonlinear Stochastic Delay Differential Equations

This paper focuses on explicit approximations for nonlinear stochastic delay differential equations (SDDEs). Under the weakly local Lipschitz and some suitable conditions, a generic truncated Euler-Maruyama (TEM) scheme for SDDEs is proposed, which numerical solutions are bounded and converge to the exact solutions in qth moment for q>0. Furthermore, the 1/2 order convergent rate is yielded. Under the Khasminskii-type condition, a more precise TEM scheme is given, which numerical solutions are exponential stable in mean square and P-1. Finally, several numerical experiments are carried out to illustrate our results.

math.NA↗

arXiv2020

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

In this article we introduce several kinds of easily implementable explicit schemes, which are amenable to Khasminski's techniques and are particularly suitable for highly nonlinear stochastic differential equations (SDEs). We show that without additional restriction conditions except those which guarantee the exact solutions possess their boundedness in expectation with respect to certain Lyapunov functions, the numerical solutions converge strongly to the exact solutions in finite-time. Moreover, based on the nonnegative semimartingale convergence theorem, positive results about the ability of explicit numerical approximation to reproduce the well-known LaSalle-type theorem of SDEs are proved here, from which we deduce the asymptotic stability of numerical solutions. Some examples and simulations are provided to support the theoretical results and to demonstrate the validity of the approach.

math.NA↗

arXiv2019

Dynamical Behaviors of the Tumor-immune System in a Stochastic Environment

This paper investigates dynamic behaviors of the tumor-immune system perturbed by environmental noise. The model describes the response of the cytotoxic T lymphocyte (CTL) to the growth of an immunogenic tumour. The main methods are stochastic Lyapunov analysis, comparison theorem for stochastic differential equations (SDEs) and strong ergodicity theorem. Firstly, we prove the existence and uniqueness of the global positive solution for the tumor-immune system. Then we go a further step to study the boundaries of moments for tumor cells and effector cells and the asymptotic behavior in the boundary equilibrium points. Furthermore, we discuss the existence and uniqueness of stationary distribution and stochastic permanence of the tumor-immune system. Finally, we give several examples and numerical simulations to verify our results.

math.PR↗