Journal
SIAM JOURNAL ON CONTROL AND OPTIMIZATION
Volume 57, Issue 6, Pages 3911-3938Publisher
SIAM PUBLICATIONS
DOI: 10.1137/18M1231833
Keywords
dynamic programming principle; fully coupled forward-backward stochastic differential equations; Hamilton-Jacobi-Bellman equation; viscosity solution
Categories
Funding
- NSF [61907022, 11671231, 11571203]
- Young Scholars Program of Shandong University [2016WLJH10]
- Natural Science Foundation of Shandong Province [ZR2019BF015]
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In this paper, we study the existence and uniqueness of viscosity solutions to a kind of Hamilton-Jacobi-Bellman (HJB) equation combined with algebra equations. This HJB equation is related to a stochastic optimal control problem for which the state equation is described by a fully coupled forward-backward stochastic differential equation (FBSDE). By extending Peng's backward semigroup approach to this problem, we obtain the dynamic programming principle and show that the value function is a viscosity solution to this HJB equation. As for the proof of the uniqueness of viscosity solution, the analysis method in Barles, Buckdahn, and Pardoux [Stochastics, 60 (1997), pp. 57-83] usually does not work for this fully coupled case. With the help of the uniqueness of the solution to FBSDEs, we propose a novel probabilistic approach to study the uniqueness of the solution to this HJB equation. We obtain that the value function is the minimum viscosity solution to this HJB equation. Especially, when the coefficients are independent of the control variable or the solution is smooth, the value function is the unique viscosity solution.
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