Hjb Equations for Certain Singularly Controlled Diffusions
نویسندگان
چکیده
over the admissible controls U . Both g and κ · u (u ∈ U) may take positive and negative values. This paper studies the corresponding dynamic programming equation (DPE), a second-order degenerate elliptic partial differential equation of HJB-type with a state constraint boundary condition. Under the controllability condition GU = R and the finiteness of H(q) = supu∈U1{−Gu · q− κ · u}, q ∈ R , where U1 = {u ∈ U : |Gu| = 1}, we show that the cost, that involves an improper integral, is well defined. We establish the following: (i) the value function for the control problem satisfies the DPE (in the viscosity sense), and (ii) the condition infq∈Rd H(q)< 0 is necessary and sufficient for uniqueness of solutions to the DPE. The existence and
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