Optimal withdrawals in a general diffusion model with control rates subject to a state-dependent upper bound

TL;DR

This paper solves a stochastic control problem involving optimal withdrawals in a diffusion process, demonstrating that the best strategy is a bang-bang control with a barrier, and provides numerical examples.

Contribution

It provides a probabilistic solution to a diffusion control problem with state-dependent bounds, identifying bang-bang strategies as optimal.

Findings

Optimal strategy is a bang-bang control with a barrier.

The solution uses a probabilistic guess-and-verify method.

Numerical examples illustrate the strategy's effectiveness.

Abstract

We consider a classical stochastic control problem in which a diffusion process is controlled by a withdrawal process up to a termination time. The objective is to maximize the expected discounted value of the withdrawals until the first-passage time below level zero. In this work, we are considering absolutely continuous control strategies in a general diffusion model. Our main contribution is a solution to the control problem under study, which is achieved by using a probabilistic guess-and-verify approach. We prove that the optimal strategy belongs to the family of bang-bang strategies, i.e. strategies in which, above an optimal barrier level, we withdraw at the highest-allowed rate, while no withdrawals are made below this barrier. Some nontrivial examples are studied numerically.