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1.	Andersson, K., et al. (author) CONVERGENCE OF A ROBUST DEEP FBSDE METHOD FOR STOCHASTIC CONTROL 2023 In: Siam Journal on Scientific Computing. - : Society for Industrial & Applied Mathematics (SIAM). - 1064-8275 .- 1095-7197. ; 45:1 Journal article (peer-reviewed)abstract In this paper, we propose a deep learning based numerical scheme for strongly coupled forward backward stochastic differential equations (FBSDEs), stemming from stochastic control. It is a modification of the deep BSDE method in which the initial value to the backward equation is not a free parameter, and with a new loss function being the weighted sum of the cost of the control problem, and a variance term which coincides with the mean squared error in the terminal condition. We show by a numerical example that a direct extension of the classical deep BSDE method to FBSDEs fails for a simple linear-quadratic control problem, and we motivate why the new method works. Under regularity and boundedness assumptions on the exact controls of time continuous and time discrete control problems, we provide an error analysis for our method. We show empirically that the method converges for three different problems, one being the one that failed for a direct extension of the deep BSDE method.

Andersson, K., et al. (author)
CONVERGENCE OF A ROBUST DEEP FBSDE METHOD FOR STOCHASTIC CONTROL
2023
In: Siam Journal on Scientific Computing. - : Society for Industrial & Applied Mathematics (SIAM). - 1064-8275 .- 1095-7197. ; 45:1
Journal article (peer-reviewed)abstract
- In this paper, we propose a deep learning based numerical scheme for strongly coupled forward backward stochastic differential equations (FBSDEs), stemming from stochastic control. It is a modification of the deep BSDE method in which the initial value to the backward equation is not a free parameter, and with a new loss function being the weighted sum of the cost of the control problem, and a variance term which coincides with the mean squared error in the terminal condition. We show by a numerical example that a direct extension of the classical deep BSDE method to FBSDEs fails for a simple linear-quadratic control problem, and we motivate why the new method works. Under regularity and boundedness assumptions on the exact controls of time continuous and time discrete control problems, we provide an error analysis for our method. We show empirically that the method converges for three different problems, one being the one that failed for a direct extension of the deep BSDE method.

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