Numerical Solutions Of High-dimensional Partial Integro-differential Equations And Forward-backward Stochastic Differential Equations Based On Deep Learning

Partial Integro differential equations(PIDEs)and forward backward stochastic differential equations with jumps(FBSDEJs)are widely used in Engineering Science,stochastic control,finance and other fields.However,with the increase of dimension,the computational complexity increases exponentially.Therefore,it is of great practical significance to study the numerical solution of high-dimensional PIDEs and FBSDEJs.Recently,deep learning methods have been proved to have good performance and accuracy for high-dimensional problems,among which backward stochastic differential equation(BSDE)method based on deep learning and deep learning reverse dynamic programming(DBDP2)method have been used to solve partial differential equations and forward and backward stochastic differential equations.In this thesis,we mainly extends the deep BSDE method and DBDP2 method to solve high-dimensional PIDEs and FBSDEJs problems,We talk about it in two parts.Firstly,we study the deep BSDE method for high-dimensional parabolic PIDEs and FBS-DEJs problems.We transform the high-dimensional PIDEs problem into the high-dimensional FBSDEJs problem by the nonlinear Feynman Kac formula.Time directization by Euler scheme.FBSDEJs is regarded as a stochastic control problem.The gradient and integral part of the solution are estimated by deep neural network,therefore,the deep BSDE scheme of FBSDEJs is constructed.The numerical results also verify the effectiveness of the algorithm,we further extended to coupled FBSDEJs,and study a posterior error estimation of the deep BSDE scheme.Then we study the DBDP2 method for solving the decoupled high-dimensional parabolic PIDEs and FBSDEJs problems.In this thesis,we estimate the solution and integral part by deep neural network,calculate the gradient of the solution by numerical diflerentialion,the approximate solution is obtained by minimizing the loss function at each time step,therefore,the DBDP2 scheme of FBSDEJs is constructed.Finally,we analyze the convergence of the scheme and get the convergence rate.