Second order parabolic Hamilton-Jacobi-Bellman equations in Hilbert spaces and stochastic control: approach

B. Goldys and Fausto Gozzi

Stochastic Processes and their Applications, 2006, vol. 116, issue 12, 1932-1963

Abstract: We study a Hamilton-Jacobi-Bellman equation related to the optimal control of a stochastic semilinear equation on a Hilbert space X. We show the existence and uniqueness of solutions to the HJB equation and prove the existence and uniqueness of feedback controls for the associated control problem via dynamic programming. The main novelty is that we look for solutions in the space L2(X,[mu]), where [mu] is an invariant measure for an associated uncontrolled process. This allows us to treat controlled systems with degenerate diffusion term that are not covered by the existing literature. In particular, we prove the existence and uniqueness of solutions and obtain the optimal feedbacks for controlled stochastic delay equations and for the first order stochastic PDE's arising in economic and financial models.

Keywords: Hamilton-Jacobi equation Stochastic evolution equation Stochastic optimal control Dynamic programming; White noise Infinite dimensions (search for similar items in EconPapers)
Date: 2006
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Citations: View citations in EconPapers (7)

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