Stability Analysis Of Numerical Methods For Several Classes Of Stochastic Delay Differential Equations

Stochastic delay differential equations (SDDEs) are widely used in engineering, physics, medical science, biology, economics and so on. It is difficult to find the analytical solutions for the overwhelming ma-jority of SDDEs, therefore, it has important theoretical and practical significance to design efficient numerical methods to solve SDDEs. In this thesis, we mainly study the stability of the numerical methods for SDDEs with Poisson jumps and the stochastic delay integro-differential equations(SDIDEs). The thesis is composed of six chapters.In the first chapter, the research background of the SDDEs and the developments of the numerical methods for the SDDEs are introduced. The main contribution of this thesis is also briefly introduced.Chapter2focuses on the almost sure exponential stability of the Euler methods for SDDEs with Poisson jumps. By using the discrete semimartingale convergence theorem, we proved that both the explicit Euler method and the backward Euler method can reproduce the almost sure exponential stability of the underlying systems provided that the stepsize is sufficiently small.In Chapter3, a class of compensated stochastic θ methods is con-structed for the nonlinear SDDEs with Poisson jumps. The mean square convergence and stability properties of the proposed methods are stud-ied. The compensated stochastic θ methods are proved to have the strong convergence rate1/2under the global Lipschitz conditions. The conditions on the stability of the analytical and numerical solutions are obtained for the complex coefficients linear scalar SDDEs with jumps. In particular, the proposed methods with1/2≤θ≤1are shown to be mean-square P-stable, which gives a natural extension of deterministic P-stability for DDEs. Chapter4is concerned with the stability of the compensated stochas-tic9methods for nonlinear SDDEs with jumps. The mean square ex-ponential stability and mean square stability conditions of the methods are obtained. It is proved that the compensated stochastic6methods with1/2≤θ≤1are mean square stable for any stepsize At=τ/m.In Chapter5, the stability properties of the stochastic9methods for nonlinear SDIDEs are considered. The sufficient conditions on mean square exponential stability of the analytical solutions and the numer-ical solutions are obtained. It is proved that the stochastic9methods with1/2≤θ≤1are mean square stable for any stepsize under the constrained grid.In the sixth chapter, an improved split-step backward Euler method is proposed to solve the nonlinear SDIDEs and the mean square expo-nential stability of the method is investigated. It is shown that the method preserves the mean-square exponential stability of the underly-ing systems with every stepsize under the constrained gridNumerical examples are given to illustrate the theory results men-tioned above.