Deep Learning Method For Solving High-dimensional Forward-backward Stochastic Differential Equations And Stochastic Optimal Control Problems

In the past decades,the numerical computation of fully-coupled forward-backward stochastic differential equations(FBSDEs in short)and stochastic optimal control problems cannot truly perform well for high-dimensional cases.Until the recent introduction of deep learning methods,the numerical computation has been greatly developed.This dissertation mainly studies the applications and computation of deep learning method in some stochastic analysis fields,including the fully-coupled FBSDEs,stochastic optimal control problem,stochastic Hamiltonian problem and non-linear expectation.The main innovation of this article is to deal with the stochastic analysis problem by constructing a suitable neural network,because the neural network is very capable of handling high-dimensional problems.For different stochastic problems,we have systematically discussed the deep learning method of stochastic problems,and propose new problem transformation and iterative algorithms.The deep learning method solves the high-dimensional calculation problems of a class of FBSDEs and stochastic optimal control problems very well,and breaks the limitation that traditional classical methods cannot deal with high-dimensional problems.There are 8 chapters in this dissertation,the first two chapter is the Introduction and Preliminaries,the third to the seventh chapters are concrete applications of the deep learning.The main contents are as follows.The first two chapters are Introduction and Preliminaries.These two chapters briefly introduce the development of stochastic analysis theory and deep learning theory,which can help readers understand the algorithms of the following chapters.We review the basic knowledge and the deep BSDE method proposed in[28].In Chapter 3,we study the numerical solutions of high-dimensional fully-coupled FBSDEs via deep learning.Here we regard the process Z as a stochastic control,and propose different neural network architectures according to different feedback forms.We systematically discuss the deep learning method under different state feedback functions.In addition,we also give the proof of the convergence under different conditions.In Chapter 4,we mainly discuss how to solve the high-dimensional stochastic optimal control problem by calculating the stochastic Hamiltonian problem obtained by the stochastic maximum principle(SMP in short).The stochastic Hamiltonian system derived from the stochastic optimal control problem is actually a coupled FBSDE with a maximum condition.In this chapter,we propose different neural network structures to solve the problem according to the different properties of the maximum condition.According to the different situations of the maximum condition,we divide the stochastic Hamiltonian system into three categories.The main advantage of using the stochastic maximum principle to solve the stochastic optimal control is that it provides a criterion for judging whether the numerical solution is close enough to the explicit solution,that is,whether the loss error is close to zero.The Chapter 5 mainly discusses a new deep learning method for solving stochastic Hamiltonian system,which is a special kind of FBSDE.For a given stochastic Hamiltonian system,we firstly look for its corresponding stochastic optimal control problem,so that the stochastic Hamiltonian system obtained by the stochastic optimal control problem through the SMP is our original given Hamiltonian system.According to the properties of stochastic Hamiltonian systems,we can divide their corresponding stochastic optimal control problems into two categories.One is that the stochastic optimal control problem can be fully expressed explicitly,and the other is not.The former is a classical stochastic optimal control problem,and the latter is a stochastic optimal control problem with a maximum condition.For the latter case,we transform it into a stochastic Stackelberg differential game,and proposed a new cross optimization method.In Chapter 6,we study the stochastic optimal control problem driven by fullycoupled FBSDEs.We firstly transform this stochastic optimal control problem into a stochastic Stackelberg differential game and solve the game by a cross optimization method.We perform more iterations on the follower’s problem to ensure that the functional of the follower is optimal.We compute the stochastic recursive utility examples in the financial market,and the results show that our cross optimization method is effective for solving the stochastic optimal control problem driven by fullycoupled FBSDEs.In Chapter 7,we briefly discuss the application of deep learning method to nonlinear expectation computation.According to the representation theorem of sublinear expectation,a sub-linear expectation can be expressed as the supremum of a family of linear expectations.At this time we can regard sub-linear expectation as the optimal value of a stochastic optimal control problem with control domain constraints.Then the methods mentioned in Chapter 4 and Chapter 5 can be used to calculate the sub-linear expectation problem,especially the high-dimensional case of sub-linear expectation problem.We compute sub-linear expectations for six different functions.The results show that for convex functions or concave functions,as well as non-negative or non-positive non-convex and non-concave functions,the deep learning network has better computational results.For non-convex and non-concave functions that can be positive or negative,the results calculated by the deep learning network are not convergent.It requires us to find a more accurate and effective method to calculate the high-dimensional sub-linear expectation problem.In last chapter,we briefly make a conclusion and discuss the work we are going to study.