Continuous viscosity solutions to linear-quadratic stochastic control problems with singular terminal state constraint

TL;DR

This paper proves the existence and uniqueness of continuous viscosity solutions for HJB equations in linear-quadratic stochastic control problems with singular terminal constraints, enabling better analysis of such control systems.

Contribution

It introduces a novel comparison principle for PDEs with singular terminal values, ensuring the continuity and uniqueness of solutions in complex control problems.

Findings

Established existence of unique nonnegative continuous viscosity solutions.

Developed a new comparison principle for PDEs with singular terminal conditions.

Validated the use of viscosity solutions for verification in control problems.

Abstract

This paper establishes the existence of a unique nonnegative continuous viscosity solution to the HJB equation associated with a Markovian linear-quadratic control problems with singular terminal state constraint and possibly unbounded cost coefficients. The existence result is based on a novel comparison principle for semi-continuous viscosity sub- and supersolutions for PDEs with singular terminal value. Continuity of the viscosity solution is enough to carry out the verification argument.