Optimal Dividend Strategies Under Markov Regime Switching Models

In this paper, we consider the optimal dividend strategies under the risk models with regime switching. Under the compound Poisson model with regime switching, we study three problems. First, with the assumption that the dividends can only be paid at a small rate, we show that the optimal strategy is the modulated threshold strategy provided that some conditions hold. In the case of two regimes and exponential claim size, we obtain an analytical solution.Second, we consider the case with unbounded dividend rate. In this situation, a lump sum of surplus may be paid as dividends, and there is a jump part of the cumulative dividends process. Besides the dividend strategy, we also consider a reinsurance strategy. The definition of viscosity solution in the presence of regime switching is given. We show that the value function can be characterized as the unique viscosity solution of the associated HJB equation. Additionally, we present a verification theorem.Third, we consider the optimal proportional reinsurance and impulse dividend strategy. Considering a class of utility functions, the object of the insurer is to select the reinsurance and dividend strategy which maximizes the expected total discounted utility of the shareholders until ruin. We study the quasi-variational inequality for this classical-impulse control problem and establish a verification theorem. We characterize the value function as the unique viscosity solution of the corresponding quasi-variational inequality.The Gamma-Omega model is discussed in the last part. Firstly, we prove that the barrier strategy is the optimal strategy. Secondly, we consider the diffusion model with regime switch-ing. In contrast to the classical risk theory, the dividends can only be paid at the arrival times of a modulated Poisson process. We modify the procedure of existing literature, and show that the optimal strategy is the modulated barrier strategy. The value function and the optimal barriers can be obtained by iteration or by solving system of differential equations. Finally, we give some comments on the results under the Gamma-Omega model with regime switching.