| 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 physics,ecology,medicine,finance,neural network and control theory since stochastic modelling describe the real world processes much better.Therefore,SFDEs have been intensively studied over the past few decades.Due to the difficulty of analysis,the study of numerical solutions for SFDEs including how to find some appropriate numerical schemes is of great,both theoretical and practical,interest.In this thesis,we study several classes of SFDEs which focus on convergence and stability of their numerical methods.This thesis is organized as follows:Chapter 1 introduces the background of stochastic differential equations(SDEs)and SFDEs.Related works on SDEs and SFDEs are summarized.Previous research on numerical analysis of SDEs and SFDEs is briefly reviewed.The contributions of this thesis are outlined.Chapter 2 investigates the convergence and stability of θ method in small moment(p ∈(0,1))for stochastic differential delay equations(SDDEs)with global Lipschitz continuous condition.Firstly,the pth moment boundedness and strong convergence ofθ method is analysed.And then,the equivalence between the pth moment exponential stability of SDDEs and that of θ method is shown.Finally,sufficient conditions for almost sure stability of SDDEs and θ method are given.Chapter 3 presents the truncated Euler-maruyama method of SFDEs under local Lipschitz condition and Khasminskii-type condition.Its pth moment boundedness is considered.We also discuss its strong convergence and show its order of Lq-convergence(2 ≤ q < p where p is the parameter in Khasminskii-type condition)under one-sided Lipschitz condition and polynomial growth condition on the drift coefficients,and global Lipschitz condition on the diffusion coefficients.Chapter 4 studies a class of highly nonlinear SDDEs with the drift coefficients f =F + F1 and the diffusion coefficients g = G + G1 under global Lipschitz condition on F1 and G1,polynomial growth condition and Khasminskii-type condition on F and Gj.We propose a partially truncated Euler-Maruyama method for SDDEs and study its pth moment boundedness(p is the parameter in Khasminskii-type condition).We also discuss its strong convergence and its order.Chapter 5,due to the fact that,for SDEs with commutative noise where drift coefficients f = F + V and diffusion coefficients g =(g1,g2,· · ·,gm)(gj=(Gj+ Uj),j = 1,2,· · ·,m),the explicit Euler method is divergent and the convergence rate of the partially truncated Euler-Maruyama method is only close to 1/2 under global Lipschitz condition on V and Ujand local Lipschitz conditions plus Khasminskii-type condition on F and Gj,we present a partially truncated Milstein method.Its pth moment boundedness and its strong convergence are considered.We show that its convergence order is close to 1 under polynomial growth condition on F and Gj.Moreover,we discuss the sufficient conditions under which the partially truncated Milstein method reproduces the mean square exponential stability. |
