Properties Of Solutions Of Stochastic Differential Equations With Non-Lipschitz Coefficient

Stochastic differential equation(SDE)can be used to describe important mathematical models in various fields(such as economy,biology,aerospace,financial product pricing,etc.).However,it is difficult to obtain the solution of SDE explicitly in many cases,so the numerical scheme of SDEs becomes more and more important.As an introduction,the first chapter introduces SDE from ordinary differential equation.At the same time,the solution of SDE is defined,and the development history of SDE numerical methods is given,mainly in the convergence and stability of numerical methods.Finally the problems to be studied are proposed.In the second chapter,we study the stochastic differential equation when the diffusion term is H?lder continuous and the drift term is local Lipschitz continuous.We derive the assumptions required for the convergence of the modified truncated EM method(MTEM),and prove that when the drift coefficient satisfies the one-side Lipschitz condition,if we find the appropriate h function,MTEM converges and the convergen-ce rate is given.For Chapter 3,a new explicit numerical scheme is proposed called the modified truncated Milstein method.It is obtained that the modified truncated Milstein scheme converges to the exact solution at an arbitrary rate close to 1 when the diffusion and drift coefficients satisfy the Khasminskii condition,and both the first derivative and second derivative satisfy local boundedness.Chapter 4 discusses the exponential stability of the modified truncated Milstein scheme,and obtains that both the discrete numerical scheme and the continuous numerical scheme exhibit asymptotic exponential stability when the diffusion and drift coefficients satisfy the local Lipschitz condition and Khasminskii condition.Chapter 5 summarizes the above results and describes future research directions.