Stochastic Delay Differential Equations, The Approximate Solution And Its Stability

In the dissertation, we consider the convergence of approximate solutions for stochastic differential delay equations and the relation of stability between approximate solutions and explicit solutions. In Chapter 1, we introduce the history and result of stability and numerical method for stochastic differential delay equations; and else, we present the structure for this dissertation. In Chapter 2, we introduce some definitions and inequalities which are always used in this dissertation. In Chapter 3, the form of stochastic differential delay equations is generalized to the neutral form, and the convergence of approximate solutions is obtained under the global Lipschitz condition; Because of the application, the convergence in probability of approximate solutions is obtained too. In Chapter 4, under the global Lipschitz condition, we investigate the relation of the stability between approximate solutions and explicit solutions for stochastic differential delay equations with jumps, and obtain the conditions for equivalence. In Chapter 5, the neutral form is embedded into the equation which appears in the Chapter 4, and we get the same condition. Differences from the literature are that: the neutral form and jumps are embedded into the equations, which is good for the produce of human.