Optimal dividend payment problems in piecewise-deterministic compound Poisson risk models

Assets are assumed to follow a diffusion process subject to some conditions. The owners can pay dividends at their discretion, but whenever assets reach zero, they have to reinvest money so that assets never go negative. With each dividend payment there is a fixed and a proportional cost, and so with reinvestments. The goal is to maximize the expected value of discounted net cash flow, i.e., dividends paid minus reinvestments. It is shown that there can be two different solutions depending on the model parameters and the costs as follows. (1) Whenever assets reach a barrier $y^*$ they are reduced to $y^*-\delta^*$ through a dividend payment, and whenever they reach 0 they are increased to $\gamma^*\leq y^*-\delta^*$ by a reinvestment. (2) There is no optimal policy, but the value function is approximated by policies of the form described in (1) for increasing barriers. We provide criteria to decide whether an optimal solution exists, and when it does not, we show how to calculate the value function. We discuss how the problem can be solved numerically and give numerical examples.

Optimal dividend policies for compound Poisson processes: The case of bounded dividend rates

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

We consider a discrete-time risk model with delayed claims and a dividend payment strategy. It is assumed that every main claim will induce a by-claim which may be delayed for one time period with a certain probability. In the evaluation of the expected present value of dividends, the interest rates are assumed to follow a Markov chain with finite state space. Dividends are paid to the shareholders according to an admissible strategy. The company controls the amount of dividends in order to maximize the cumulative expected discounted dividends prior to ruin minus the expected discounted penalty value at ruin. We obtain some properties of the optimal dividend-payment strategy, and offer high efficiency algorithms for obtaining the optimal strategy, the optimal value function and the expectation of time of ruin under the optimal strategy. Our method is mainly to transform the value function. Numerical examples are presented to illustrate the transformation method and the impact of the penalty for ruin on the optimal strategy, the corresponding expected present value of dividends and the expectation of time of ruin.

In this note, we consider the optimal dividend payment strategy in regime-switching model under the bounded dividend rate restriction. The surplus of the insurance company is modelled as a regime switching Brownian motion, which has the drift and volatility from two-states Markov process control. The objective is to find the dividend policy which maximizes the expected total discounted dividend pay-outs until the time of bankruptcy. We derive the Hamilton-Jacobi-Bellman equation and closed form solution for the problem. Finally, a numerical simulation is presented to characterize the behavior of the optimal dividend payment strategy.

We consider the optimal stopping problem, based on the compound binomial model with capital injection controlled by optimal dividend strategy. First, the value function V(x) is established. Second, via the Bellman equation satisfied by the value function V(x) which maximizes the discounted value of define between dividend payment and the penalized discounted capital injection, we find the optimal stopping time τ*. Furthermore an optimal stopping model with capital injection in order to maximize the shareholders’ interests is obtained.

For spectrally negative Lévy risk processes we consider a general version of de Finetti's optimal dividend problem in which the ruin time is replaced with a general drawdown time from the running maximum in its value function. We identify a condition under which a barrier dividend strategy is optimal among all admissible strategies if the underlying process does not belong to a small class of compound Poisson processes with drift, for which the take-the-money-and-run dividend strategy is optimal. It generalizes the previous results on dividend optimization from ruin time based to drawdown time based. The associated drawdown functions are discussed in detail for examples of spectrally negative Lévy processes.

This paper proposes and studies an optimal dividend problem in which a two-state regime-switching environment affects the dynamics of the company's cash surplus and, as a novel feature, also the bankruptcy level. The aim is to maximize the total expected profits from dividends until bankruptcy. The company's optimal dividend payout is therefore influenced by four factors simultaneously: Brownian fluctuations in the cash surplus, as well as regime changes in drift, volatility and bankruptcy levels. In particular, the average profitability can assume different signs in the two regimes. We find a rich structure of the optimal strategy, which, depending on the interaction of the model's parameters, can be either of barrier-type or of liquidation-barrier type. Furthermore, we provide explicit expressions of the optimal policies and value functions. Finally, we complement our theoretical results by a detailed numerical study, where also a thorough analysis of the sensitivities of the optimal dividend policy with respect to the problem's parameters is performed.

In the dual risk model, we consider the optimal dividend and capital injection problem, which involves a random time horizon and a ruin penalty. Both fixed and proportional costs from the transactions of capital injection are considered. The objective is to maximize the total value of the expected discounted dividends, and the penalized discounted both capital injections and ruin penalty during the horizon, which is described by the minimum of the time of ruin and an exponential random variable. The explicit solutions for optimal strategy and value function are obtained, when the income jumps follow a hyper-exponential distribution. Besides, some numerical examples are presented to illustrate our results.

This paper studies the optimal debt ratio and dividend strategy for an insurer under the model with the coefficients depending on the state of the economy. The object is to maximize the total expected discounted utility of dividend payment of the insurer. The optimal strategy and value function are characterized by the classical solution of the associated Hamilton–Jacobi–Bellman equation which can be reduced to a system of nonlinear PDEs. Considering logarithmic and power utility, we show the existence of classical solution to the system by the ordered upper-lower solution method, and verify that the solution is indeed the value function.

On optimal dividend payments and related problems

In this paper, we consider an insurance company which has two dependent lines of business. Each business is modeled by a compound Poisson risk process. Using the diffusion approximation theory, we get an approximated diffusion model. The company applies a dynamic reinsurance policy to reduce the ruin probability and pays dividends to keep competitive. The objective is to find out the value function and the optimal policy which maximizes the expected cumulative discounted dividend. Since the excess-of-loss reinsurance is more profitable than the proportional reinsurance, we discuss the optimal excess-of-loss reinsurance and dividend problem in this approximate diffusion model. The problem is considered in two situations: The dividend rate is bounded and the dividend rate is unbounded. Since the risks are dependent, five cases need to be dealt with when the dividend rate is bounded and two cases need to be investigated when the dividend rate is unbounded. In both situations, explicit expressions for the value function and the corresponding optimal strategies are obtained.

On the bailout dividend problem with periodic dividend payments for spectrally negative Markov additive processes

Optimal reinsurance and dividend for a diffusion model with capital injection: Variance premium principle

We consider a optimal dividend and stopping problem for two-dimensional compound Poisson risk model. The two companies allow to help each other by transffering payments. The goal of the companies is to maximize the expected cumulative discounted dividends payments up to the time of ruin. The problem is formulated as a singular control and optimal stopping problem. We explicitly construct the value function and the optimal strategy when claims are exponentially distributed and present two numerical examples.