Numerical Methods For Solving Two Kinds Of Stochastic Delay Differential Equations

Stochastic delay differential equation as an important mathematical model of nat-ural phenomenon and social phenomenon with uncertainty and influenced by certain history information, has been widely used in life science, economics, environmental science, mechanical engineering, control, etc. According to the type of noise in the system, the stochastic delay differential equations can be classified into stochastic delay differential equations driven by Brownian motion (SDDEs) and by the frac-tional Brownian motion (FSDDEs). Mostly, the analytical solutions are not known, and one has to resort to numerical techniques. Therefore, the development of nu-merical methods for stochastic delay differential equations has become meaningful and valuable both in theory and in applications.This paper focuses on proposing numerical methods for nonlinear SDDEs and FSDDEs.Firstly, tamed Euler schemes and balanced schemes are established for nonlinear stochastic delay differential equations:The convergence and stability analysis of these two numerical schemes are given. Furthermore, the theoretical results are verified and efficiency of numerical methods is shown by numerical examples.Secondly, for stochastic delay differential equations driven by a fractional Brow-nian motion (FSDDEs) with Hurst parameter 1/2< H< 1, we construct Euler scheme and modified Euler scheme, then prove that the both proposed schemes are conver-gent respectively in the sense of LP norm for FSDDEs. Furthermore, we give con-vergence orders of the two schemes. Compared to the classical Euler scheme, the modified Euler scheme can obtain higher accuracy.In the end, some concluding remarks are given to show what can be improved in the paper and some future work.