Stochastic control problems and HJB equations with excluded parameters of random inputs

TL;DR

This paper develops a novel class of second order stochastic HJB equations that incorporate observable but unpredictable parameters, reducing state space complexity in optimal control problems with unknown parameter dynamics.

Contribution

It introduces a new HJB framework that excludes explicit parameter specifications, enabling dimension reduction and handling control-dependent diffusion in nonlinear equations under Cordes conditions.

Findings

Formulation of second order stochastic HJB equations with excluded parameters

Reduction of state space dimension in control problems

Handling of control-dependent diffusion coefficients

Abstract

This paper introduces a new type of second order stochastic backward Hamilton-Jacobi-Bellman (HJB) equations for optimal stochastic control problems with a currently observable but non-predicable parameter process, in addition to the driving Brownian motion. The main feature of this HJB equation is that it excludes specifications of the parameter process which dynamics can be unspecified or unknown. This allows to reduce the dimension of the state space. The paper considers the case of control dependent diffusion coefficients and fully nonlinear HJB equations under so-called Cordes conditions.