Search

In this work, we propose two classes of two-step Milstein-type schemes : the double-implicit Milstein scheme and the split two-step Milstein scheme, to solve stochastic differential equations (SDEs). Our results reveal that the two new schemes are strong convergent with order one. Moreover, with a restriction on stepsize, these two schemes can preserve the exponential mean square stability of the original SDEs, and the decay rate of numerical solution will converge to the decay rate of the exact solution. Numerical experiments are performed to confirm our theoretic findings. [ABSTRACT FROM AUTHOR]

In this paper, we first prove the strong convergence of the split-step theta methods for non-autonomous stochastic differential equations under a linear growth condition on the diffusion coefficient and a one-sided Lipschitz condition on the drift coefficient. Then, if the drift coefficient satisfies a polynomial growth condition, we further get the rate of convergence. Finally, the obtained results are supported by numerical experiments. [ABSTRACT FROM PUBLISHER]

This paper examines exponential mean square stability of the split-step theta approximation and the stochastic theta method for the stochastic differential delay equations and stochastic ordinary differential equations (SODEs) under a coupled monotone condition on drift and diffusion coefficients. It is shown that forthe two classes of the theta approximations can preserve the exponential mean square stability when some conditions on the stepsize and drift coefficient are imposed, but for, without the globally Lipschitz continuity, these two classes of theta methods show exponentially mean square stability unconditionally. Moreover, for sufficiently small stepsize, the decay rate as measured by the bound of the Lyapunov exponent can be reproduced arbitrarily accurately. Some results in this paper extend the existing results for linear SODEs to nonlinear stochastic differential equations (SDEs), and also improve our previous results of numerical stability of nonlinear SDEs. [ABSTRACT FROM AUTHOR]

Recently, several scholars discussed the question of under what conditions numerical solutions can reproduce exponential stability of exact solutions to stochastic delay differential equations, and some delay-independent stability criteria were obtained. This paper is concerned with delay-dependent stability of numerical solutions. Under a delay-dependent condition for the stability of the exact solution, it is proved that the backward Euler method is mean-square exponentially stable for all positive stepsizes. Numerical experiments are given to confirm the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the numerical stability of linear stochastic delay integro-differential equations (SDIDEs). A sufficient condition for mean square stability of the exact solution of a linear SDIDE with multiplicative noise is derived. Then the mean square stability of stochastic θ-methods is investigated, and it is shown that the numerical solution can reproduce the mean square stability of the exact solution under appropriate conditions. At last, we present some numerical experiments to support our conclusions. [ABSTRACT FROM AUTHOR]