Optimal dividend problems for Sparre Andersen risk model with bounded dividend rates

ABSTRACT

This paper concerns the optimal dividend problem with bounded dividend rate for Sparre Andersen risk model. The analytic characterizations of admissible strategies and Markov strategies are given. We use the measure-valued generator theory to derive a measure-valued dynamic programming equation. The value function is proved to be of locally finite variation along the path, which belongs to the domain of the measure-valued generator. The verification theorem is proved without additional assumptions on the regularity of the value function. Actually, the value function may have jumps. Under certain conditions, the optimal strategy is presented as a Markov strategy with space-time band structure. We present an iterative algorithm to approximate the optimal value function and the optimal dividend strategy. As applications, some numerical examples are given.  

Optimal dividend strategy with transaction costs for an upward jump model

In this paper, we consider the optimal dividend problem with transaction costs when the incomes of a company can be described by an upward jump model. Both fixed and proportional costs are considered in the problem. The value function is defined as the expected total discounted dividends up to the time of ruin. Although the same problem has already been studied in the pure diffusion model and the spectrally negative Lévy process, the optimal dividend problem in an upward jump model has two different aspects in determining the optimal dividends barrier and in the property of the value function. First, the value function is twice continuous differentiable in the diffusion case, but it is not in the jump model. Second, under the spectrally negative Lévy process, downward jumps will not cause any payment actions; however, it might trigger dividend payments when there are upward jumps. In deriving the optimal barriers, we show that the value function is bounded by a linear function. Using this property, we establish the verification theorem for the value function. By solving the quasi-variational inequalities associated with this problem, we obtain the closed-form solution to the value function and hence the optimal dividend strategy when the income sizes follow a common exponential distribution. In the presence of a fixed transaction cost, it is shown that the optimal strategy is a two-barrier policy, and the optimal barriers are only dependent on the fixed cost and not the proportional cost. A numerical example is used to illustrate how the fixed cost plays a significant role in the optimal dividend strategy and also the value function. Moreover, an increased fixed cost results in larger but less frequent dividend payments.