Optimal Impulse Control for Growth-Restricted Linear Diffusions with Regime Switching

Abstract

This paper investigates the impulse cash-flow withdrawal (dividend distribution or consumption) optimization problem for a broad class of growth restricted linear diffusions with drift and volatility dependent on both the level of surplus and the environment/economy regime (described by a Markov chain). There are proportional and fixed costs associated with each cash-flow withdrawal transaction, and the objective of the optimization problem is to determine the optimal policy that maximizes the expected total cumulative discounted cash-flow withdrawals (dividend payments/consumption) minus expenses. This work extends all the existing work on the same problem under the diffusion setting by extending the underlying dynamics to a much more general class of Markov modulated diffusions and unifies the existing work on optimal impulse dividend control. We show that when there are two regimes, the value function, as a function of the initial cash reservoir and the initial environmental state, can be determined by solving a set of second order ordinary differential equations. We further show that either the optimal strategy is a regime switching lump-sum dividend barrier strategy or there exists no optimal strategies.