Optimal dividend distribution under Markov regime switching

In this paper we investigate the problem of optimal dividend distribution in the presence of regime shifts. We consider a company whose cumulative net revenues evolve as a drifted Brownian motion modulated by a finite state Markov chain, and model the discount rate as a deterministic function of the current state of the chain. The objective is to maximize the expected cumulative discounted dividend payments until the moment of bankruptcy, which occurs the first time that the cash reserves (the cumulative net revenues minus cumulative dividend payments) hit zero. We show that, if the drift is positive in each regime, it is optimal to adopt a barrier strategy at certain positive regime-dependent levels, and explicitly characterize the value function as the fixed point of a contraction. In the case that the drift is small and negative in some regime, the optimal strategy takes a different form, which we explicitly identify in the case that there are two regimes. We also provide a numerical illustration of the sensitivities of the optimal barriers and the influence of regime-switching. Research supported by EPSRC grant EP/D039053/1. The research was partially carried out while the authors were based at King’s College London. This paper is based on Chapter 3 of the first author’s PhD thesis (submitted in April 2008). Acknowledgements: We thank A. Dassios, R. Norberg, M. Zervos, and participants of the 12th International Congress of IME (Insurance: Mathematics and Economics) in Dalian, China, and the Bachelier Conference in London (July 2008) for useful suggestions. School of Finance, Nanjing University of Finance and Economics, Nanjing, 210046, China, email: [email protected] Department of Mathematics, Imperial College London, London SW7 2AZ, UK, email: [email protected]