Strong Convergence Of Explicit Numerical Methods For Two Types Of Nonlinear Stochastic Differential Equations

With the continuous advancement of scientific research,nonlinear stochastic differential equations are widely used in control engineering,biology,finance and other fields to describe the world more accurately.However,since the analytical expressions of nonlinear stochastic differential equations are difficult to obtain,the application of numerical methods or approximation techniques to numerically approximate the exact solution of nonlinear stochastic differential equations has both important theoretical significance and extensive application value.This thesis mainly studies the strong convergence of the truncated EM method for two types of stochastic differential equations.This paper mainly consists of the following chapters:Chapter 1 briefly introduces the research background and some numerical methods of stochastic differential equations,and then Chapter 2 briefly introduces the basic symbols,basic concepts and common inequalities needed in this paper and so on.Chapter 3 investigates the explicit positivity preserving numerical approximations for the Cox-Ingersoll-Ross(CIR)model driven by fractional Brown motion with Hurst parameter ?(1/2,1).Firstly,for overcoming the difficulties caused by the unbounded diffusion coefficient,an auxiliary equation with constant diffusion coefficient obtained by proper Lamperti transformation is used.Then,by means of Malliavin calculus,we show that the truncated EM scheme applied to this auxiliary equation not only ensures the positivity of the numerical solution,but also has the -order rate of the root mean square error over a finite time interval.Moreover,by transforming back,an explicit scheme for the original CIR model is obtained and has the same convergence order.Finally,an example with some simulations is provided to support the theoretical results and demonstrate the validity of the approach.Chapter 4 mainly studies the truncated EM method for generalized nonlinear stochastic Volterra integro-differential equations.Firstly,under local Lipschitz condition plus the Khasminskii-type condition,the th moment boundedness and strong convergence of the truncated EM numerical solution are proved.Furthermore,the convergence rates of the truncated EM method of the numerical solutions close to 1/2 is obtained under some stronger assumptions.Finally,a numerical example is given to illustrate the feasibility and validity of our theoretical results.Chapter 5 summarizes the main work of this paper and puts forward further research ideas.