Stochastic Hamilton–Jacobi–Bellman Equations

This paper studies the following form of nonlinear stochastic partial differential equation: \[ {gathered} - dΦ _t = {inf }v ∈ U \{ {1/2∑i,j {[ {σ σ ^ * } ]ᵢⱼ (x,v,t)} ∂ x_i x_j Φ _t (x) + ∑_i {b_i (x,v,t)} ∂ x_i Φ _t (x) + L(x,v,t)} . \\ . { + ∑i,j {σᵢⱼ(x,v,t)∂ x_i Ψ j,t (x)} } \}dt - Ψ _t (x)dW_t , Φ _T (x) = h(x), \\ {gathered}\] where the coefficients σ ᵢⱼ, bᵢ, L, and the final datum h may be random. The problem is to find an adapted pair (Φ ,Ψ )(x,t) uniquely solving the equation. The classical Hamilton–Jacobi–Bellman (HJB) equation can be regarded as a special case of the above problem. An existence and uniqueness theorem is obtained for the case where σ does not contain the control variable v. An optimal control interpretation is given. The linear quadratic case is discussed as well.