EM Approximations To Several Types Of Stochastic Functional Differential Equations

In practice,systems depend on not only the current state,but also the past states as well as they are subject to noise disturbance.Stochastic functional differential equations(SFDEs)have been used in many branches of science such as ecology,medicine,finance,neural network and control theory since stochastic modelling describe the real world process much better.Same as Stochastic differential equations,stochastic functional differential equations do have unique solutions under the Local Lipsthitz condition plus linear growth condition,but true solution are not available under general condition.Therefore,the study of numerical solution for SFDEs including how to find some appropriate numerical schemes is great,both theoretical and practical interest.First of all,we talk about the EM approximation to a class of SFDEs.Numerical approximation by the Euler-Maruyama method with random variable step-size is defined here,and we prove that the numerical approximation converge to zero almost surely.Furthermore,we study a more particular kind of SFDEs that neutral stochastic functional differential equations(NSFDEs).we also prove that the numerical approximation defined by the Euler-Maruyama method with random variable step-size converge to zero almost surely.Finally we use truncated EM method to study the convergence rates of the numerical solutions for neutral stochastic delay differential equations(NSDDEs)with time-dependent delay under 1/2.