# Stochastic control on the half-line and applications to the optimal dividend/consumption problem

## TL;DR

This paper studies a stochastic control problem with barrier constraints, proves smooth solutions to the associated Hamilton-Jacobi-Bellman equations, and applies these results to optimize dividend and consumption strategies.

## Contribution

It introduces a method to solve barrier-constrained stochastic control problems and applies it to optimal dividend and consumption issues.

## Key findings

- Proved existence of smooth solutions to Hamilton-Jacobi-Bellman equations under general conditions.
- Developed a fixed point approach using stochastic representations for linear equations.
- Applied the theoretical results to derive optimal strategies in dividend and consumption problems.

## Abstract

We consider a stochastic control problem with the assumption that the system is controlled until the state process breaks the fixed barrier. Assuming some general conditions, it is proved that the resulting Hamilton Jacobi Bellman equations has smooth solution. The aforementioned result is used to solve the optimal dividend and consumption problem. In the proof we use a fixed point type argument, with an operator which is based on the stochastic representation for a linear equation.