Special Session 59: 

AI Based Simulation and Analysis by Malliavin Calculus for Backward Stochastic Partial Differential Equations

Wanyang Dai
Nanjing University
Peoples Rep of China
We develop a generic numerical scheme to simulate the paired adapted strong solution (i.e., a pair of vector random fields) to a generalized Cauchy (terminal-value) problem, i.e., a unified system of backward stochastic partial differential equations (B-SPDEs including infinite-dimensional B-SDEs) driven by Brownian motions. This unified system can be used to model many real-world system dynamics such as optimal control and differential game problems. To better understand our numerical scheme and mathematical regime in the well-known artificial intelligence (AI) convolutional neural network (CNN), we use a CNN to model our numerical process by conditional expectation projection. It consists two convolution parts: W layers of backward networks and L layers of reinforcement iterations. Furthermore, it is a completely discrete and iterative algorithm in terms of both time and space with almost sure (a.s.) convergence supported by both theoretical proof and numerical examples. In doing so, the system is assumed to be high-dimensional and vector-valued and whose drift and diffusion coefficients may involve nonlinear and high-order partial differential operators. Under general local Lipschitz and linear growth conditions, the unique existence of the paired adapted strong solution to the system is proved by constructing a suitable Banach space to handle the difficulty that the partial differential orders on both sides of these equations are different. During the proof, we also develop new techniques of random field Malliavin calculus to show the unique existence of paired adapted strong solutions to two embedded systems of the first and second Malliavin derivative based B-SPDEs under random environments.