| Stochastic Delay Differential Equations (SDDEs) can be regarded as extension of certain model problems for Delay Differential Equations (DDEs) which consider the random factor. It can also be regarded as the extension of uncertain model problems for Stochastic Ordinary Differential Equations (SODEs) considering the delay factor. Thus SDDEs can always simulate the scientific problem in practical truthfully. They have been widely used in Physics, Chemistry, Cybernetics, Finance, Neural Networks, Bionomics, etc. For the research method of SDDEs, it can neither be treated as DDEs nor SODEs, so it produces difficulties. The same as the DDEs and SDDEs, to obtain the theoretical solution of SDDEs will be very hard, hence it is important to study the numerical methods on solving the SDDEs.The study of numerical methods for solving SDDEs is underway at present, the research at home and abroad are mainly focused on linear SDDEs. In this paper, the main object is to extend the previous research results to nonlinear cases. Secondly, the delay termτis the multiple of stepsize h to dispose the problem, we approximate the delay term with linear interpolation to break this restriction. We study some kinds of numerical methods for solving nonlinear SDDEs, and obtain a series of stable and convergent results. These results can be regarded as the extension of the known results in present papers. The main results of this thesis are as follows:(1) First, we approximate the delay term with linear interpolation through the values of known neighbor points, this method is a new attempt corresponding with the known method for dealing with the delay term, which always use the technique entirely for treating the delay termτis the integer times of the stepsize h. For a kind of initial problems of nonlinear SDDEs, we then present the convergent results of Euler-Maruyama methods with linear interpolation procedure in the mean of mean-square. They are the extension of related results in previous papers.(2) The MS-stability and GMS-stability, which is the mean square concept of numerical methods for SDDEs, was extended from linear experiment equations to general nonlinear case. For one dimensional Initial Value Problem(IVP) of nonlinear SDDE, when the problem itself satisfies the sufficient condition that the zero solution is mean square asymptotically stable, we prove that when the drift term satisfies some given restrictions, the Euler-Maruyama methods is MS-stable and the Euler-Maruyama methods with linear interpolation procedure is GMS-stable.(3) We extend the Euler-Maruyama methods to a much more general class of semi-implicit methods, i.e. stochasticθ-methods. For the IVP of nonlinear SDDEs in one dimension, when the problem itself satisfies the sufficient condition that the zero solution is mean square asymptotically stable, we prove that when the drift term satisfies some given restrictions, the semi-implicit Euler method is MS-stable and the semi-implicit Euler method with linear interpolation procedure is GMS-stable.(4) Consider solving a class of very important equation, Fokker-Planck equations. For one dimensional Fokker-Planck equations of white noise driven by stochastic system, here we prove that the Milstein methods is MS-stable and the Milstein methods with linear interpolation procedure is GMS-stable for solving these problems when the problem itself satisfy the sufficient condition that the zero solution is mean square asymptotically stable.(5) When the diffusion term equals to zero, nonlinear SDDEs are degenerated to deterministic problems. This chapter is concerned with the error analysis of one-leg methods when applied to nonlinear stiff DDEs with a variable delay. It is proved that a one-leg methods with Lagrangian linear interpolation procedure is D-convergent of oder p if and only if it is A-stable and consistent of order p in the classical sense for ODEs. The results obtained can be regarded as extension of that for DDEs with constant delay presented in documents.
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