Numerical Methods For Several Kinds Of Stochastic Differential Equations

Stochastic differential equations play an important role in many fields such as physics,finance,control,and biology,etc.Since stochastic differential equations can simulate various stochastic problems well,the researches on its theories and applications have attracted more and more people’s attention.However,it is usually difficult to obtain the explicit expression of the exact solution of stochastic differential equations,applying appropriate numerical algorithms to simulate stochastic differential equations has both great theoretical significance and extensive practical value.This paper mainly studies the convergence and stability of several kinds of stochastic differential equations,mainly including the following aspects:For stochastic differential equations with random variable stepsize,we give the choice of stepsize under the θ-Maruyama and θ-Milstein methods,and prove that theθ methods can reproduce the almost sure stability of exact solutions of stochastic differential equations by means of the non-negative discrete semi-martingale convergence theorem and its martingale difference form.For the semi-linear stochastic evolution equation driven by Riemann-Liouville fractional Brownian motion with Hurst constant H<1/2,we first prove the pth moment exponential stability of mild solution.Then,based on the maximal inequality,the uniform boundedness of pth moment of both exact and numerical solutions exponential Euler method are studied,and the strong convergence of the exponential Euler method is established.Finally,two multi-dimensional examples are carried out to demonstrate the consistency with theoretical results.For stochastic differential equations driven by time-changed Brownian motion which contain two drift terms,one driven by the random time change Et,and the other driven by a regular,non-random time variable t.Strong convergence and convergence rate in the finite time of the Euler-Maruyama method on the particle system is discussed.Numerical example illustrates that "particle corruption" will not occur on the whole particle system and show the consistency with convergence result.