Smooth solutions to discounted reward control problems with unbounded discount rate and financial applications

TL;DR

This paper addresses continuous-time stochastic control problems with unbounded discount rates, establishing conditions for smooth solutions to the HJB equation and exploring financial applications like consumption-investment models.

Contribution

It introduces general assumptions ensuring smooth solutions to HJB equations with unbounded discount rates and discusses extensions to dynamic games and financial economics.

Findings

Existence of smooth classical solutions under new assumptions

Verification methods for control problems with unbounded discount rates

Application to consumption-investment problems in finance

Abstract

We consider a discounted reward control problem in continuous time stochastic environment where the discount rate might be an unbounded function of the control process. We provide a set of general assumptions to ensure that there exists a smooth classical solution to the corresponding HJB equation. Moreover, some verification reasoning are provided and the possible extension to dynamic games is discussed. At the end of the paper consumption - investment problems arising in financial economics are considered.