Stochastic (Functional) Differential Equations With Irregular Coefficients And Their Related Problems

This dissertation mainly contains two parts.The first one is concerned with numerical analysis of stochastic differential equations(SDEs)with non-Lipschitz coefficients(H(?)lder continuity).The second one is about the stationary distribution of the stochastic Lotka-Volterra model under non-Lipschitz condition.The whole dissertation consists of the following five chapters:Chapter 1 briefly introduces some backgrounds of SDEs with irregular coefficients.Moreover,we will introduce some results of the existence and uniqueness of the solutions of SDEs whose diffusion coefficient is only H(?)lder continuous.Some numerical methods for these equations are also introduced in this chapter.In the last part of this chapter,we will also introduce the stochastic Lotka-Volterra model with irregular coefficients which has been studied a little and give the contribution of this dissertation.Chapter 2 examines the unique strong solution and its numerical simulation of SDEs with H(?)lder character.The first part concerns that the diffusion coefficients consist of H(?)lder con-tinuous part and local H(?)lder continuous part.We consider their strong solution and the strong L~1convergence of the Euler method.The second part concerns that the diffusion coefficients only consist of local H(?)lder continuous part which could be Fisher-Wright model exactly.We also prove the existence and uniqueness of the strong solution and the strong L~1convergence of the Euler method.Chapter 3 analyzes the convergence rate of the Truncated-Euler method for SDEs with H(?)lder diffusion coefficient.If the drift coefficient satisfies local Lipschitz and monotone lin-ear growth condition,we can prove the strong L~1convergence of the Truncated-Euler method.Moreover,the convergence rate is same as that of the classical Euler method.The strong L~q(q?2)convergence and the convergence rate are also given in this chapter.Chapter 4 considers the stationary distribution of stochastic functional differential equa-tions with non-lipschitz coefficients.According to the project mapping,we can change the original functional equation into a degenerate equation without delay.If the noise is weak,we can prove the new equation has the Feller property and the stationary distribution exists.If the noise is strong,we can even obtain the strong Feller property of the new equation by examin-ing the H(?)rmander condition.Finally,we can prove the existence of the stationary distribution and it is absolutely continuous with Lebesgue measure.Chapter 5 examines the ergodicity of stochastic functional differential equation(SFDEs)with infinite delay.As an extension of the stationary distribution of SFDEs in Chapter 4,we generally consider the existence and uniqueness of the stationary distribution on the space C_r.Some useful results including the existence of the unique strong solution and pth moment uniformly bounded are given.Under some dissipative conditions,we show that there admits a unique invariant measure of the solution mapping x_twith nonlinear distribution delay in the drift term.The convergence rate to the invariant measure is exponential under the Wasserstein distance.