Recently, the deep learning method has been used for solving forward backward stochastic differential equations (FBSDEs) and parabolic partial differential equations (PDEs). It has good accuracy and performance for high-dimensional problems. In this paper, we mainly solve fully coupled FBSDEs through deep learning and provide three algorithms. Several numerical results show remarkable performance especially for high-dimensional cases.

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We discuss a new type of fully coupled forward-backward stochastic differential equations (FBSDEs) whose coefficients depend on the states of the solution processes as well as their expected values, and we call them fully coupled mean-field forward-backward stochastic differential equations (mean-field FBSDEs). We first prove the existence and the uniqueness theorem of such mean-field FBSDEs under some certain monotonicity conditions and show the continuity property of the solutions with respect to the parameters. Then we discuss the stochastic optimal control problems of mean-field FBSDEs. The stochastic maximum principles are derived and the related mean-field linear quadratic optimal control problems are also discussed.

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Abstract-In this paper, we develop acceleration strategies for option pricing with non-linear Backward Stochastic Differential Equation (BSDE), which appears as a robust and valuable tool in financial markets. An efficient binomial lattice based method is adopted to solve the BSDE numerically. In order to reduce the global memory access frequency, the kernel invocation is avoided to be performed on each time step. Furthermore, for evaluating the affect of load balance to the performance, we provide two different acceleration strategies and compare them with running time experiments. The acceleration algorithms exhibit tremendous speedup over the sequential CPU implementation and therefore suitable for real-time application.