Convergence rate of numerical solutions to SFDEs with jumps

In this paper, we are interested in numerical solutions of stochastic functional differential equations with jumps. Under a global Lipschitz condition, we show that the pth-moment convergence of Euler–Maruyama numerical solutions to stochastic functional differential equations with jumps has order 1/p for any p ≥ 2. This is significantly different from the case of stochastic functional differential equations without jumps, where the order is 1/2 for any p ≥ 2. It is therefore best to use the mean-square convergence for stochastic functional differential equations with jumps. Moreover, under a local Lipschitz condition, we reveal that the order of mean-square convergence is close to 1/2, provided that local Lipschitz constants, valid on balls of radius j, do not grow faster than log j

An averaging principle for neutral stochastic functional differential equations driven by Poisson random measure

In this paper, we study the averaging principle for neutral stochastic functional differential equations (SFDEs) with Poisson random measure. By stochastic inequality, Burkholder-Davis-Gundy’s inequality and Kunita’s inequality, we prove that the solution of the averaged neutral SFDEs with Poisson random measure converges to that of the standard one in (Formula presented.) sense and also in probability. Some illustrative examples are presented to demonstrate this theory

Mini-Workshop: Dynamics of Stochastic Systems and their Approximation

The aim of this workshop was to bring together specialists in the area of stochastic dynamical systems and stochastic numerical analysis to exchange their ideas about the state of the art of approximations of stochastic dynamics. Here approximations are considered in the analytical sense in terms of deriving reduced dynamical systems, which are less complex, as well as in the numerical sense via appropriate simulation methods. The main theme is concerned with the efficient treatment of stochastic dynamical systems via both approaches assuming that ideas and methods from one ansatz may prove beneficial for the other. A particular goal was to systematically identify open problems and challenges in this area

Convergence Rate of EM Scheme for SDDEs

In this paper we investigate the convergence rate of Euler-Maruyama scheme for a class of stochastic differential delay equations, where the corresponding coefficients may be highly nonlinear with respect to the delay variables. In particular, we reveal that the convergence rate of Euler-Maruyama scheme is 1/2$ for the Brownian motion case, while show that it is best to use the mean-square convergence for the pure jump case, and that the order of mean-square convergence is close to 1/2.Comment: Page 1

Convergence Rate of Numerical Solutions for Nonlinear Stochastic Pantograph Equations with Markovian Switching and Jumps

The sufficient conditions of existence and uniqueness of the solutions for nonlinear stochastic pantograph equations with Markovian switching and jumps are given. It is proved that Euler-Maruyama scheme for nonlinear stochastic pantograph equations with Markovian switching and Brownian motion is of convergence with strong order 1/2. For nonlinear stochastic pantograph equations with Markovian switching and pure jumps, it is best to use the mean-square convergence, and the order of mean-square convergence is close to 1/2