On Optimal Dividends in the Dual Model

Abstract

We revisit the dividend payment problem in the dual model of Avanzi et al. ([3], [2], and [4]).Using the fluctuation theory of spectrally positive Lévy processes, we give a short exposition in which we show the optimality of barrier strategies for all such Lévy processes. Moreover, we characterize the optimal barrier using the functional inverse of a scale function. We also consider the capital injection problem of [4] and show that its value function has a very similar form to the one in which the horizon is the time of ruin.

Optimal Dividends and Capital Injections in the Dual Model with a Random Time Horizon

“…By the fluctuation theory of Lévy processes, we find the optimal strategy and the optimal return function. If the random time horizon goes to infinity, and if the fixed transaction costs on capital injection tend to zero, the two suboptimal problems in this paper become those in [3], respectively. Furthermore, when the positive jumps of the Lévy process are hyper-exponential compound Poisson jumps, we can derive the results in [10].…”

“…In [1,2], the authors studied how the expectation of the discounted dividends until ruin can be calculated in the dual compound Poisson risk model. Recently, in [3][4][5], the optimal dividend problems were studied in a general spectrally positive Lévy risk model.…”

“…In [10], Avanzi et al discussed the same problem in the dual compound Poisson model with diffusion. In [3], the optimal dividend and capital injection problem was discussed for a general spectrally positive Lévy process. In addition, transaction cost, which usually includes two partsproportional cost and fixed cost-, is an important factor in business activities.…”

“…Unlike [3], we assume that the ruin may occur even under the rescue of capital injection, since transaction costs are also considered here. Moreover, we assume that the surplus process is killed by an independent exponential random time as in [12].…”

Optimal Capital Structure With Scale Effects Under Spectrally Negative Lévy Models

“…A number of authors have succeeded in extending the classical results to the spectrally negative Lévy model by way of scale functions. We refer the reader to [5,6] for stochastic games, [4,7,8,25,33] for the optimal dividend problem, [1,3] for American and Russian options, and [15,17,27,32] for credit risk. In particular, Egami and Yamazaki [18] considered a general optimal stopping problem for spectrally negative Lévy processes and obtained the first-order condition for maximization over threshold strategies; the results are also confirmed numerically by [40] in a multiple stopping setting.…”