2018/05/07 by Mingshang Hu, Hu, Mingshang, Shaolin Ji +3
Decision Sciences · Economics, Econometrics and Finance · #35K15 #60H10 #93E20 #Economic theories and models #FOS: Mathematics #Optimization and Control (math.OC) #Risk and Portfolio Optimization #Stochastic processes and financial applications
paper · pdf · doi:10.48550/arxiv.1805.02337
openalex publication_date 2018/05/07 · openalex created_date 2019/06/27 · openalex updated_date 2026/07/28
In this paper, we study the existence and uniqueness of viscosity solutions to a kind of Hamilton-Jacobi-Bellman (HJB) equations 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. 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 Baeles-BP 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.