Viscosity Solutions of Stochastic Hamilton--Jacobi--Bellman Equations
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In this paper we study the fully nonlinear stochastic Hamilton-Jacobi-Bellman (HJB) equation for the optimal stochastic control problem of stochastic differential equations with random coefficients. The notion of viscosity solution is introduced, and we prove that the value function of the optimal stochastic control problem is the maximal viscosity solution of the associated stochastic HJB equation. For the superparabolic cases when the diffusion coefficients are deterministic functions of time, states and controls, the uniqueness is addressed as well.
24 pages
viscosity solution, Viscosity solutions to Hamilton-Jacobi equations in optimal control and differential games, Probability (math.PR), Dynamic programming in optimal control and differential games, Optimality conditions for problems involving randomness, optimal stochastic control, Viscosity solutions to PDEs, backward stochastic partial differential equation, Mathematics - Analysis of PDEs, Stochastic partial differential equations (aspects of stochastic analysis), Optimization and Control (math.OC), stochastic Hamilton-Jacobi-Bellman equation, FOS: Mathematics, Optimal stochastic control, Mathematics - Optimization and Control, Mathematics - Probability, Analysis of PDEs (math.AP)
viscosity solution, Viscosity solutions to Hamilton-Jacobi equations in optimal control and differential games, Probability (math.PR), Dynamic programming in optimal control and differential games, Optimality conditions for problems involving randomness, optimal stochastic control, Viscosity solutions to PDEs, backward stochastic partial differential equation, Mathematics - Analysis of PDEs, Stochastic partial differential equations (aspects of stochastic analysis), Optimization and Control (math.OC), stochastic Hamilton-Jacobi-Bellman equation, FOS: Mathematics, Optimal stochastic control, Mathematics - Optimization and Control, Mathematics - Probability, Analysis of PDEs (math.AP)
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