Numerical Analysis Of Stochastic Differential Equations And Stochastic Stabilization

This dissertation mainly contains two parts, the first one is concerned with numerical analysis of nonlinear stochastic differential equations (SDEs) with non-global Lipschitz co-efficients and addresses the two questions:(1) If the SDE is stable, whether the numerical approximation can reproduce this stability?(2) If the numerical method inherits the stability of the SDE, whether the Lyapunov exponent of exact solutions can be preserved as the stepsize tends to zero? The second part examines the influences of the stochastic factors (Brownian mo-tion, Poisson process and Markov chain) on stability and stabilization in the senses of almost sure and pth-moment stability. The whole dissertation consists of the following eight chapters:Chapter1briefly introduces some traditional applications and the research on the numer-ical analysis of stochastic differential equations and stochastic stabilization. The contribution of this dissertation is also listed in this chapter.Chapter2examines the strong convergence and exponential mean square stability of the split-step theta-Euler (SSTE) approximation and the stochastic linear theta-Euler (SLTE) method for nonlinear SODEs with non-global Lipschitz coefficients. Under the one-sided Lip-schitz and a polynomial Lipschitz conditions on drift as well as global Lipschitz condition on diffusion, the strong convergece rate and exponential mean square stability are obtained for the SSTE and SLTE schemes. Moreover, for sufficiently small stepsize, the decay rate as measured by the bound of the Lyapunov exponent can be reproduced arbitrarily accurately.Chapter3analyzes the two classes of theta-Milstein schemes:split-step theta-Milstein (SSTM) scheme and stochastic linear theta-Milstein (SLTM) scheme for stochastic differen-tial equations (SDEs) with non-global Lipschitz continuous coefficients. Under the similar conditions on the coefficients in Chapter2, the two classes of theta-Milstein schemes converge strongly to the exact solution with the order1. The exponential mean square stability and ex-ponential decay rates of the two classes of theta-Milstein schemes are also investigated in this chapter.Chapter4proposes a semi-tamed Euler (STE) scheme to approximate the SODEs with the drift coefficient satisfying the one-sided Lipschitz condition and equipped with the Lipschitz continuous part and non-Lipschitz continuous part. The STE scheme is explicit and converges strongly with the standard order one-half to the exact solution of the SODE. This chapter also proves that STE scheme can reproduce the exponential mean square stability and Lyapunov exponent bound of the exact solution.Chapter5studies SSTE and SLTE methods for the stochastic differential delay equation-s (SDDEs) and obtains the strong convergence rate under non-global Lipschitz coefficients. Under a coupled monotone condition on drift and diffusion coefficients, the exponential mean square stability is also examined. The similar convergence and stability as that in Chapter2for SODEs are obtained for SDDEs.Chapter6examines the exponential mean square stability of the exact and numerical so-lutions for neutral stochastic differential delay equations (NSDDEs). We first get the improved exponential mean square stability criterion for the exact solution. Then we investigate the ex-ponential mean square stability of the SLTE and SSTE schemes, and obtain the stability for NSDDEs similar to SDDEs in Chapter5.Chapter7focuses on regime-switching jump diffusion, which includes three classes of random processes:Brownian motions, Poisson processes, and Markovian chains. This chapter examines the influences of these processes on stability and stabilization in the senses of almost sure and pth-moment stability. First, a scalar linear system is treated as a benchmark case. Then stabilization of systems with one-sided linear growth is considered. Finally, nonlinear systems that have a finite explosion time are treated, in which regularization (suppressing the noise) and stabilization are achieved by introducing appropriate diffusions together with Poisson and Markov chain perturbations.Chapter8studies stability and stochastic stabilization of numerical solutions of a class of regime-switching jump diffusion systems. For the linear scalar system, the pth moment and almost sure exponential stability of the EM approximation are investigated. For the nonlinear system, this chapter investigates the almost sure exponential stability of EM and BEM method under the different growth condition on drift. These results show that all these three classes of stochastic factors may serve as stabilizing factors and play positive roles for the stability property of both exact and numerical solutions.