In this paper we consider the optimal dividend problem for an insurance company whose risk process evolves as a spectrally negative Lévy process in the absence of dividend payments. The classical dividend problem for an insurance company consists in finding a dividend payment policy that maximizes the total expected discounted dividends. Related is the problem where we impose the restriction that ruin be prevented: the beneficiaries of the dividends must then keep the insurance company solvent by bail-out loans. Drawing on the fluctuation theory of spectrally negative Lévy processes we give an explicit analytical description of the optimal strategy in the set of barrier strategies and the corresponding value function, for either of the problems. Subsequently we investigate when the dividend policy that is optimal amongst all admissible ones takes the form of a barrier strategy.

The current paper presents a short survey of stochastic models of risk control and dividend optimization techniques for a financial corporation. While being close to consumption /investment models of Mathematical Finance, dividend optimization models possess special features which do not allow them to be treated as a particular case of consumption/investment models.

We investigate a model of a corporation which faces constant liability payments and which can choose a production/business policy from an available set of control policies with different expected profits and risks. The objective is to maximize the expected present value of the total dividend distributions. The main purpose of this paper is to deal with the impact of constraints on business activities such as inability to completely eliminate risk (even at the expense of reducing the potential profit to zero) or when such a risk cannot exceed a certain level. We analyze the case in which there is no restriction on the dividend pay-out rates. By delicate analysis on the corresponding HamiltonJacobi -Bellman equation we compute explicitly the optimal return function and determine the optimal policy.

This paper represents a model for a financial valuation of a firm which has control on its risk as well as potential profit by choosing different business activities among those available to it. Furthermore the firm has an option of investing its reserve in a financial market consisting of a risk free asset (Bond) and a risky asset (Stock). The example we consider is that of a large corporation such as an insurance company, whose liquid assets in the absence of control and investments fluctuate as a Brownian motion with a constant positive drift and a constant diffusion coefficient. We interpret the diffusion coefficient as risk exposure, while drift is associated with potential profit. At each moment of time there is an option to reduce risk exposure, simultaneously reducing the potential profit, like using proportional reinsurance with another carrier for an insurance company. The company invests its reserve in a financial market, which is described by a classical BlackScholes model....

Abstract. We consider an optimal control problem of a property insurance company with proportional reinsurance strategy. The insurance business brings in catastrophe risk, such as earthquake and flood. The catastrophe risk could be partly reduced by reinsurance. The management of the company controls the reinsurance rate and dividend payments process to maximize the expected present value of the dividends before bankruptcy. This is the first time to consider the catastrophe risk in property insurance model, which is more realistic. We establish the solution of the problem by the mixed singularregular control of jump diffusions. We first derive the optimal retention ratio, the optimal dividend payments level, the optimal return function and the optimal control strategy of the property insurance company, then the impacts of the catastrophe risk and key model parameters on the optimal return function and the optimal control strategy of the company are discussed. MSC(2000): Primary91B30,91B70,91B28;Secondary60H10,60H30.

We study a model of a corporation which has the possibility to choose various production/business policies with different expected profits and risks. In the model there are restrictions on the dividend distribution rates as well as restrictions on the risk the company can undertake. The objective is to maximize the expected present value of the total dividend distributions. We outline the corresponding Hamilton–Jacobi–Bellman equation, compute explicitly the optimal return function and determine the optimal policy. As a consequence of these results, the way the dividend rate and business constraints affect the optimal policy is revealed. In particular, we show that under certain relationships between the constraints and the exogenous parameters of the random processes that govern the returns, some business activities might be redundant, that is, under the optimal policy they will never be used in any scenario. 1. Introduction. In