Theoretical And Numerical Analysis Of Several Kinds Of Stochastic Equations With Memory

In order to investigate some complicated systems with memory and uncertainty,the stochastic equation model with memory has been established and has become a research hotspot in recent years.The obvious difference from the stochastic equation without memory is that the solution of the stochastic equation with memory is no longer a Markov process.In commonly,stochastic equations with memory mainly include:stochastic Volterra integral equations,stochastic Volterra integro-differential equations,stochastic fractional differential equations,stochastic fractional integro-differential equations,stochastic delay differential equations and so on.This dissertation studies several kinds of stochastic equations with memory as follows:In Chapter 2,for stochastic fractional integro-differential equations with regular kernels,we investigate their well-posedness as well as the strong convergence of the Euler–Maruyama(EM)method based on the connection between stochastic fractional integro-differential equations and stochastic Volterra integral equations.Specifically,under the non-Lipschitz and linear growth conditions,the existence,uniqueness and continuous dependence on the initial value of the true solution will be derived in detail.Under the same conditions as the well-posedness,the strong convergence of the EM method will also be proven.Furthermore,when the non-Lipschitz condition is replaced with the global Lipschitz condition,the mean-square convergence rate of the EM method is also obtained.In particular,when the order??(2/1,1]of the Caputo fractional derivative,the EM method can achieve strong first-order superconvergence.In Chapter 3,for stochastic fractional integro-differential equations with Abel-type weakly singular kernels,under the local Lipschitz continuous and linear growth conditions,both the well-posedness of nonlinear initial value problems and the strong convergence of EM methods are studied.Furthermore,under the global Lipschitz and linear growth conditions,the mean-square convergence rate of the EM method is also studied.In particular,when the Caputo fractional derivative degenerates to the integer-order derivative,the EM method can attain strong first-order superconvergence,which actually improves the corresponding result obtained by[J.Comput.Appl.Math.383(2021)113156].In Chapter 4,for Lévy-driven stochastic Volterra integral equations with doubly singular kernels,under the global Lipschitz continuous and linear growth conditions,we are devoted to analyzing the existence and uniqueness of analytical solutions as well as the mean-square convergence rate of the EM method.In addition,we will also develop a fast EM method by making use of the efficient sum-of-exponentials approximation,which improves the computational efficiency of the EM method.In Chapter 5,for the linear case of overdamped generalized Langevin equations with fractional noise,we study the mean-square convergence rate of the EM method,which will solve the problem left by[ESAIM Math.Model.Numer.Anal.54(2020)431–463].In particular,the obtained result explains in detail how the mean-square convergence rate of the EM method depends on the degree of weakly singular kernels and the Hurst index of the fractional Brownian motion.In Chapter 6,for numerically solving stochastic delay differential equations,we develop the discontinuous Galerkin(DG)method via the Wong–Zakai approximation.When the drift coefficient satisfies the global Lipschitz continuous and linear growth conditions,we will prove that the DG method is strongly convergent for the additive noise case.Under the condition that the drift coefficient is the one-sided Lipschitz continuous,we will also prove that the DG method is stable for the additive noise case.