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

2024/06/17 by Guérin, Hélène, Mata, Dante, Renaud, Jean-François +1
#60J60 #60J70 #93E20 #FOS: Mathematics #Optimization and Control (math.OC) #Probability (math.PR)

paper · doi:10.48550/arxiv.2406.12067

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.

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