Optimal stochastic control with recursive cost functionals of stochastic differential systems reflected in a domain∗,∗∗
School of Mathematics and Statistics, Shandong University, Weihai,
2 Institute of Mathematics and Department of Finance and Control Sciences, School of Mathematical Sciences, Fudan University, Shanghai 200433, P.R. China
Received: 21 October 2013
Revised: 25 August 2014
The paper is concerned with optimal control of a stochastic differential system reflected in a domain. The cost functional is implicitly defined via a generalized backward stochastic differential equation developed by Pardoux and Zhang [Probab. Theory Relat. Fields 110 (1998) 535–558]. The value function is shown to be the unique viscosity solution to the associated Hamilton–Jacobi–Bellman equation, which is a fully nonlinear parabolic partial differential equation with a nonlinear Neumann boundary condition. The proof requires new estimates for the reflected stochastic differential system.
Mathematics Subject Classification: 60H99 / 60H30 / 35J60 / 93E05 / 90C39
Key words: Hamilton–Jacobi–Bellman equation / nonlinear Neumann boundary / value function / backward stochastic differential equations / dynamic programming principle / viscosity solution
Juan Li has been supported by the NSF of P.R. China (Nos. 11071144, 11171187, 11222110), Shandong Province (Nos. BS2011SF010, JQ201202), SRF for ROCS (SEM), Program for New Century Excellent Talents in University (No. NCET-12-0331), 111 Project (No. B12023).
Shanjian Tang is supported in part by the National Natural Science Foundation of China (Grants #10325101 and #11171076), by Science and Technology Commission, Shanghai Municipality (Grant No. 14XD1400400), by Basic Research Program of China (973 Program) Grant #2007CB814904, by the Science Foundation of the Ministry of Education of China Grant #200900071110001, and by WCU (World Class University) Program through the Korea Science and Engineering Foundation funded by the Ministry of Education, Science and Technology (R31-20007).
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