Strong Convergence And Stability Of Numerical Methods For Stochastic Differential Equations With Non-globally Lipschitz Continuous Coefficients

Stochastic differential equations(SDEs) have extremely extensive applications in many fields such as economics, biology, medicine, fiance and engineering. However, many SDEs arising in these applications cannot be solved analytically, hence one needs to develop ef-fective numerical methods to solve them. We first propose the so called split-step (θ1,θ2,θ3) method and the high-order split-step (θ1,θ2,θ3) method which contain many classical meth-ods such as Euler method, Backward Euler method, Split-step backward Euler method, Split-step θ method, Split-step one-leg θ method, Stochastic θ method, Milstein method, Stochastic θ-Milstein method and Drifting Split-step Backward Milstein method (DSSBM). Under non-globally Lipschitz continuous condition, we investigate the strong convergence and stability of the split-step (θ1,θ2,θ3) method and the high-order split-step(θ1,θ2,θz) method. The whole dissertation contains the following six parts:In Chapter1, we first outline some applications of stochastic differential equations, s-tochastic delay differential equations and stochastic differential equations with jumps. Then, we review the present state of strong convergence of numerical methods, some basic con-ceptions and inequalities used in the numerical analysis of stochastic problems. Finally, the main work is listed.In Chapter2, we propose the split-step (θ1,θ2,θ3) method and prove the strong con-vergence of this method for non-autonomous stochastic differential equations with the drift coefficient satisfying one-sided Lipshctiz condition and the diffusion coefficient satisfying global lipschitz condition. Our results show that when02>1/2, the split-step (θ1,θ2,θ3) method is strong convergent. Then, if the drift coefficient satisfies a polynomial growth condition, we further obtain that the rate of convergence is of order one. Meanwhile this method is mean-square stable. Furthermore, we extend these results to SDEs with jumps.In Chapter3, the strong convergence of the compensated split-step (θ1,θ2,θ2) method is investigated under a weaker conditions. More precisely, the diffusion and the drift coef-ficients are both locally Lipschitz continuous and the jump-diffusion coefficient is globally Lipschitz continuous, while they all satisfy the monotone condition. These conditions admit especially that the diffusion coefficient can be highly nonlinear.In Chapter4, we extend the split-step (θ1,θ2,θ2) method to solve stochastic delay dif- ferential equations. We prove the strong convergence of this method under a local Lipschitz condition and a coupled condition on the drift and diffusion coefficients. In particular, these conditions allow that the diffusion coefficient can be highly nonlinear.In Chapter5, based on the previously proposed split-step (θ1,θ2,θ3) method, we first propose the high-order split-step (θ1,θ2，θ3) method for non-autonomous SDEs driven by non-commutative noise. Then, we prove that this method is convergent with strong order of one for SDEs with the drift coefficient satisfying a polynomial growth condition and a one-sided Lipschitz continuous condition and the diffusion coefficient satisfying global lipschitz condition.In Chapter6, we further consider the mean-square stability of the high-order split-step (θ1,θ2，θ3) method. First, we obtain the linear mean square stability and nonlinear exponen-tial mean square stability of this method under some limitations of meshes. Furthermore, we investigate the mean square stability of the split-step (θ1,θ2,θ2) method. When θ2>3/2, this method is unconditionally mean square stable for liner SDEs with real coefficients. Under some limitations of meshes, we obtain the exponential mean-square stability of the high-order split-step (θi,θ2,θ2)method for the nonlinear non-autonomous SDEs with the drift and diffusion coefficients satisfying a weaker coupled condition.