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Presented By: Financial/Actuarial Mathematics Seminar - Department of Mathematics

Convergence of the Backward Deep BSDE Method with Applications to Optimal Stopping Problems.

ZImu Zhu of UCSB

The optimal stopping problem is one of the core problems in financial markets, with broad appli- cations such as pricing American and Bermudan options. The deep BSDE method [Han, Jentzen and E, PNAS, 115(34):8505-8510, 2018] has shown great power in solving high-dimensional forward- backward stochastic differential equations (FBSDEs), and in- spired many applications. However, the method solves backward stochastic differential equations (BSDEs) in a forward manner, which can not be used for optimal stopping prob- lems that in general require running BSDE backwardly. To overcome this difficulty, a recent paper [Wang, Chen, Sudjianto, Liu and Shen, arXiv:1807.06622, 2018] proposed the back- ward deep BSDE method to solve the optimal stopping problem. In this paper, we provide the rigorous theory for the backward deep BSDE method. Specifically, 1. We derive the a posteriori error estimation, i.e., the error of the numerical solution can be bounded by the training loss function; and; 2. We give an upper bound of the loss function, which can be suffi- ciently small subject to universal approximations. We give two numerical examples, which present consistent performance with the proved theory. This is a joint work with C.Gao, S.Gao and R.Hu.

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