Strong Convergence Analysis Of Numerical Methods For Two Classes Of Stochastic Differential Equations

The theory, computation and applications of stochastic (delay) differential equations (SDEs or SDDEs) have been extensively discussed in the recent decades. By exploration and study on physics, medicine, dynamics, ecology, biology and other fields, many stochastic models with fractional derivatives can more really describe the objective laws of nature. These models are not only related to current state, some previous states or the entire history of the state, but also related to the derivatives of the state variable at previous time. They are called stochastic fractional differential equations (SFDEs) and neutral stochastic delay differential equations (NSDDEs). Most of NSDDEs and SFDEs cannot be analytically solved, so numerical algorithms have become fairly important. Now, some researchers have discussed numerical methods for NSDDEs with the drift and diffusion coefficients satisfying the global (or local) Lipschitz and linear growth conditions. And, only few encouraging results of solving numerically SFDEs have been reported in the existing literature. The existence of uncertainty, neutrality and fractional derivatives brings some challenges in solving such problems. Therefore, it is very necessary to solve numerically such problems. The main contents are depicted as follows.In chapter 1, we briefly introduce the research background, research current situation and previous relative results for NSDDEs and SFDEs. And, we state the main works of this paper. Finally, the used symbols are explained.In chapter 2, we elaborate some basic knowledge of stochastic analysis and fractional calculus theory so as to the follow-up study.In chapter 3, we propose the split-step theta (SST) method for NSDDEs. The strong convergence of the SST method with θ ∈[0, 1] is proved for NSDDEs, where the correspond-ing coefficients may be highly nonlinear with respect to the delay variables. Moreover, we obtain the strong convergence order with 1/2. A numerical example shows that the obtained theoretical results are correct.In chapter 4, firstly, we propose the Euler-Maruyama (EM) method for SFDEs. Second-1y, the strong convergence of the EM method is obtained for NSDDEs with α∈1/2 (1/2,1], where the drift and diffusion coefficients satisfy the global Lipschitz and linear growth conditions. Moreover, we obtain the strong convergence order of α- 1/2. Finally, a numerical example confirms that the obtained theoretical results are effective.