The Numerical Algorithms And Theory For Several Classes Of Stochastic Functional Differential Equations

Random fluctuations are abundant in natural or engineered systems. Therefore, stochastic modelling has come to play an important role in various fields like biology, me-chanics, economics, medicine and engineering. Moreover, these systems are sometimes sub-ject to memory effects, when their time evolution depends on their past history with noise disturbance. Stochastic functional differential equations (SFDEs) are often used to model such systems. They can be regarded as a generalization of both deterministic functional differential equations (FDEs) and stochastic ordinary delay differential equations (SODEs). Explicit solutions of SFDEs can rarely be obtained. Thus, it has become an important issue to develop numerical methods for SFDEs.This thesis is devoted to the investigation of numerical algorithms for several classes of stochastic functional differential equations. The stability, convergence and computational implementation of these methods are analyzed. In particular, a multi-scale approach that can reduce the computational burden is constructed on the basis of the predictor-corrector method given previously. The construction of the thesis is as follows.Chapter 2 presents a strong predictor-corrector method for the numerical solution of stochastic delay differential equations (SDDEs) of Ito-type. The method is proved to be mean-square convergent of order min(1/2.p) under the Lipschitz condition and the linear growth condition, where p is the exponent of Holder condition of the initial function. A stability criterion for this type of method is derived. It is shown that for certain choices of the flexible parameter p the derived method can have a better stability property than more commonly used numerical methods. Numerical results are reported confirming con-vergence properties and comparing stability properties of methods with different parameters p. Finally, the vectorised simulation is discussed and it is shown that this implementation is much more efficient. Chapter 3 deals with almost sure and moment exponential stability of a class of predictor-corrector methods applied to the stochastic differential equations of Ito-type. Sta-bility criteria for this type of methods are derived. The methods are shown to maintain almost sure and moment exponential stability for all sufficiently small timesteps under ap-propriate conditions. Numerical experiment further testifies these theoretical results.In chapter 5, a new explicit stochastic scheme of strong order 1 is proposed for stochas-tic delay differential equations (SDDEs) with sufficiently smooth drift and diffusion coeffi-cients and a scalar Wiener process. The method is derivative-free and is shown to be stable in mean square. A stability theorem for the continuous strong approximation of the solution of a linear test equation by the Milstein method is also proved, which shows the stability bound is larger than bounds given previously in the literature. The case of linear SDDEs is further investigated, in order to compare the stability bounds of these two methods. Numer-ical experiments are given to confirm the stability properties obtained.Chapter 6 presents a multi-scale approach for simulating time-delay biochemical reaction systems when there are wide ranges of molecular numbers. We construct a new approach that can reduce the computational burden on the basis of the idea of a partitioned system and recent developments with stochastic simulation algorithm and the delay stochastic simulation method. It is shown that this algorithm is much more efficient than existing methods such as DSSA method and the modified next reaction method. Some numerical results arc reported, confirming the accuracy and computational efficiency of the approximation.