Class Of Stochastic Delay Differential Equations Numerical Methods, Convergence And Stability,

Stochastic Delay Differential Equations (SDDEs) can be regarded as extension of certain problems for Delay Differential Equations (DDEs) which consider the random factor. It can also be regarded as the extension of uncertain problems for Stochastic Ordinary Differential Equations (SODEs) considering the delay factor. Therefore, SDDEs can always simulate truthfully certain scientific problem in practical. They have been widely used in Physics, Chemistry, Control Theory, Finance, Biology and other fields.In recent years, some scholars have begun to research convergence and stability of numerical methods for Stiff Stochasic Differential Equations (SSDEs) and proposed some very effective implicit numerical methods for solving SSDEs. The stiff stochastic problems can be regarded as extension of deterministic stiff problems. In the deterministic problems, in order to ensure stability, we generally use implicit numerical methods for solving stiff problems. Thus, it is very necessary with implicit numerical methods to solve stiff stochastic problems.Balance method is very effective implicit numerical method for solving stiff stochastic differential equations. It is different with some scholars who have studied implicit numerical methods——the introduction of implicitness is restricted to the deterministic terms. The balance method are fully implicit numerical method for involving both deterministic implicit terms (or drift-implicit terms) and stochastic implicit terms. As far as we know, there is no study on convergence and stability of balance method for solving SDDEs and Neutral Stochastic Pantograph Differential Equations (NSPDEs). Therefore, it is meaningful to investigate these problems. In this paper, we researh convergence and stability of balaced method for solving SDDEs and balanced semi-implicit Euler method for solving Stochastic Pantograph Differential Equations (SPDEs), the convergence of balanced semi-implicit Euler method for solving NSPDEs. Our main results in the thesis are as follows: (1) We investigate convergence and stability of balaced method for solving SDDEs. It is proved that the balanced method is mean-square convergence with stong order 1/2. Moreover, we give mean-square stability contidion of the balanced method for Linear Stochastic Delay Differential Equations (LSDDEs).(2) We propose for solving Neutral Stochastic Functional Differential Equations(NSFDEs) of the implicit numerical method——a balanced semi-implicit Euler method, and deal with convergence of the balanced semi-implicit Euler method for solving NSPDEs. It is proved that the balanced semi-implicit Euler method is mean-square convergence with stong order 1/2. We will be the conclusion of Theorem 3.2.2 in this text applied to SPDEs and be available to the convergence of conclusion that the balanced semi-implicit Euler method for SPDEs is mean-square convrgence with stong order 1/2. The conclusions are also new.(3) We investigate mean-square stability of the balanced semi-implicit Euler method for solving Linear Stochastic Pantograph Differential Equations (LSPDEs). We give mean-square stability contidion of the balanced semi-implicit Euler method for LSPDEs.