Optimal Dividend Policy When Cash Reserves Follow a Jump-Diffusion Process Under Markov-Regime Switching

We study the problem of optimal dividend payments for a company of limited liability whose cash reserves in the absence of dividends follow a Markov-modulated jump-diffusion process with positive drifts, where parameters and the discount rate are modulated by a finite-state irreducible Markov chain and governed by a deterministic function of the current state of the chain. In this setup, our main purpose is to maximize the expected cumulative discounted dividend payments until the moment of bankruptcy, the first time that cash reserves are nonpositive. Along the lines of Jiang and Pistorius \cite [23], we extend their results to our setup by using fixed point theorem and stochastic control to prove that it is also optimal to adopt a modulated barrier strategy at certain positive regime-dependent levels and that value function can be explicitly characterized as the fixed point of a contraction