Journal
IEEE TRANSACTIONS ON AUTOMATIC CONTROL
Volume 66, Issue 5, Pages 2319-2325Publisher
IEEE-INST ELECTRICAL ELECTRONICS ENGINEERS INC
DOI: 10.1109/TAC.2020.3004717
Keywords
Optimal control; Viscosity; Indexes; Differential equations; Stochastic processes; Dynamic programming; Europe; Backward stochastic differential equations (BSDE); Hamilton– Jacobi– Bellman (HJB) equations; risk-sensitive optimal control; viscosity solutions
Funding
- National Research Foundation of Korea - Ministry of Science and ICT, South Korea [NRF-2017R1E1A1A03070936, NRF-2017R1A5A1015311]
- Institute for Information & communications Technology Promotion (IITP) - Korea government (MSIT), South Korea [2018-0-00958]
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In this article, a generalized risk-sensitive optimal control problem is studied with the objective functional defined by a controlled BSDE. The dynamic programming principle and viscosity solution for the value function are obtained, with applications to the risk-sensitive European option pricing problem.
In this article, we consider the generalized risk-sensitive optimal control problem, where the objective functional is defined by the controlled backward stochastic differential equation (BSDE) with quadratic growth coefficient. We extend the earlier results of the risk-sensitive optimal control problem to the case of the objective functional given by the controlled BSDE. Note that the risk-neutral stochastic optimal control problem corresponds to the BSDE objective functional with linear growth coefficient, which can be viewed as a special case of the article. We obtain the generalized risk-sensitive dynamic programming principle for the value function via the backward semigroup associated with the BSDE. Then we show that the corresponding value function is a viscosity solution to the Hamilton-Jacobi-Bellman equation. Under an additional parameter condition, the viscosity solution is unique, which implies that the solution characterizes the value function. We apply the theoretical results to the risk-sensitive European option pricing problem.
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