Hamilton Jacobi Bellman equations in infinite dimensions with quadratic and superquadratic Hamiltonian

TL;DR

This paper studies Hamilton-Jacobi-Bellman equations in infinite-dimensional spaces with quadratic and superquadratic Hamiltonians, enabling analysis of stochastic control problems involving Ornstein-Uhlenbeck processes with unbounded controls.

Contribution

It introduces methods to solve infinite-dimensional HJB equations with complex Hamiltonians, extending stochastic control theory to unbounded control processes.

Findings

Established existence of solutions for infinite-dimensional HJB equations.

Applied results to stochastic control problems with Ornstein-Uhlenbeck processes.

Extended the class of Hamiltonians for which HJB equations can be analyzed.

Abstract

We consider Hamilton Jacobi Bellman equations in an inifinite dimensional Hilbert space, with quadratic (respectively superquadratic) hamiltonian and with continuous (respectively lipschitz continuous) final conditions. This allows to study stochastic optimal control problems for suitable controlled Ornstein Uhlenbeck process with unbounded control processes.