Optimal Dividend Payments for the Piecewise-Deterministic Poisson Risk Model

This paper considers the optimal dividend payment problem in piecewise-deterministic compound Poisson risk models. The objective is to maximize the expected discounted dividend payout up to the time of ruin. We provide a comparative study in this general framework of both restricted and unrestricted payment schemes, which were only previously treated separately in certain special cases of risk models in the literature. In the case of restricted payment scheme, the value function is shown to be a classical solution of the corresponding HJB equation, which in turn leads to an optimal restricted payment policy known as the threshold strategy. In the case of unrestricted payment scheme, by solving the associated integro-differential quasi-variational inequality, we obtain the value function as well as an optimal unrestricted dividend payment scheme known as the barrier strategy. When claim sizes are exponentially distributed, we provide easily verifiable conditions under which the threshold and barrier strategies are optimal restricted and unrestricted dividend payment policies, respectively. The main results are illustrated with several examples, including a new example concerning regressive growth rates.Comment: Key Words: Piecewise-deterministic compound Poisson model, optimal stochastic control, HJB equation, quasi-variational inequality, threshold strategy, barrier strateg

On the Optimal Dividend Problem for Insurance Risk Models with Surplus-Dependent Premiums

This paper concerns an optimal dividend distribution problem for an insurance company with surplus-dependent premium. In the absence of dividend payments, such a risk process is a particular case of so-called piecewise deterministic Markov processes. The control mechanism chooses the size of dividend payments. The objective consists in maximazing the sum of the expected cumulative discounted dividend payments received until the time of ruin and a penalty payment at the time of ruin, which is an increasing function of the size of the shortfall at ruin. A complete solution is presented to the corresponding stochastic control problem. We identify the associated Hamilton-Jacobi-Bellman equation and find necessary and sufficient conditions for optimality of a single dividend-band strategy, in terms of particular Gerber-Shiu functions. A number of concrete examples are analyzed

Numerical methods for optimal dividend payment and investment strategies of regime-switching jump diffusion models with capital injections

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"Itô's Lemma" and the Bellman equation: An applied view

Rare and randomly occurring events are important features of the economic world. In continuous time they can easily be modeled by Poisson processes. Analyzing optimal behavior in such a setup requires the appropriate version of the change of variables formula and the Hamilton-Jacobi-Bellman equation. This paper provides examples for the application of both tools in economic modeling. It accompanies the proofs in Sennewald (2005), who shows, under milder conditions than before, that the Hamilton-Jacobi-Bellman equation is both a necessary and sufficient criterion for optimality. The main example here consists of a consumption-investment problem with labor income. It is shown how the Hamilton-Jacobi-Bellman equation can be used to derive both a Keynes-Ramsey rule and a closed form solution. We also provide a new result. --Stochastic differential equation,Poisson process,Bellman equation,Portfolio optimization,Consump

On a periodic dividend barrier strategy in the dual model with continuous monitoring of solvency

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