On the Bail-Out Optimal Dividend Problem

This paper studies the optimal dividend problem with capital injection under the constraint that the cumulative dividend strategy is absolutely continuous. We consider an open problem of the general spectrally negative case and derive the optimal solution explicitly using the fluctuation identities of the refracted-reflected L\'evy process. The optimal strategy as well as the value function are concisely written in terms of the scale function. Numerical results are also provided to confirm the analytical conclusions.Comment: To appear in Journal of Optimization Theory and Applications. Keywords: stochastic control, scale functions, refracted-reflected L\'evy processes, bail-out dividend proble

Optimal debt ratio and dividend payment strategies with reinsurance

This paper derives the optimal debt ratio and dividend payment strategies for an insurance company. Taking into account the impact of reinsurance policies and claims from the credit derivatives, the surplus process is stochastic that is jointly determined by the reinsurance strategies, debt levels, and unanticipated shocks. The objective is to maximize the total expected discounted utility of dividend payment until financial ruin. Using dynamic programming principle, the value function is the solution of a second-order nonlinear Hamilton–Jacobi–Bellman equation. The subsolution–supersolution method is used to verify the existence of classical solutions of the Hamilton–Jacobi–Bellman equation. The explicit solution of the value function is derived and the corresponding optimal debt ratio and dividend payment strategies are obtained in some special cases. An example is provided to illustrate the methodologies and some interesting economic insights.postprin

Numerical Solutions of Optimal Risk Control and Dividend Optimization Policies under A Generalized Singular Control Formulation

This paper develops numerical methods for finding optimal dividend pay-out and reinsurance policies. A generalized singular control formulation of surplus and discounted payoff function are introduced, where the surplus is modeled by a regime-switching process subject to both regular and singular controls. To approximate the value function and optimal controls, Markov chain approximation techniques are used to construct a discrete-time controlled Markov chain with two components. The proofs of the convergence of the approximation sequence to the surplus process and the value function are given. Examples of proportional and excess-of-loss reinsurance are presented to illustrate the applicability of the numerical methods.Comment: Key words: Singular control, dividend policy, Markov chain approximation, numerical method, reinsurance, regime switchin

Error estimates of penalty schemes for quasi-variational inequalities arising from impulse control problems

This paper proposes penalty schemes for a class of weakly coupled systems of Hamilton-Jacobi-Bellman quasi-variational inequalities (HJBQVIs) arising from stochastic hybrid control problems of regime-switching models with both continuous and impulse controls. We show that the solutions of the penalized equations converge monotonically to those of the HJBQVIs. We further establish that the schemes are half-order accurate for HJBQVIs with Lipschitz coefficients, and first-order accurate for equations with more regular coefficients. Moreover, we construct the action regions and optimal impulse controls based on the error estimates and the penalized solutions. The penalty schemes and convergence results are then extended to HJBQVIs with possibly negative impulse costs. We also demonstrate the convergence of monotone discretizations of the penalized equations, and establish that policy iteration applied to the discrete equation is monotonically convergent with an arbitrary initial guess in an infinite dimensional setting. Numerical examples for infinite-horizon optimal switching problems are presented to illustrate the effectiveness of the penalty schemes over the conventional direct control scheme.Comment: Accepted for publication in SIAM Journal on Control and Optimizatio

Classical and Impulse Control for the Optimization of Dividend and Proportional Reinsurance Policies with Regime Switching

We consider the optimal proportional reinsurance and dividend strategy. The surplus process is modeled by the classical compound Poisson risk model with regime switching. Considering a class of utility functions, the object of the insurer is to select the reinsurance and dividend strategy that maximizes the expected total discounted utility of the shareholders until ruin. By adapting the techniques and methods of stochastic control, we study the quasi-variational inequality for this classical and impulse control problem and establish a verification theorem. We show that the optimal value function is characterized as the unique viscosity solution of the corresponding quasi-variational inequality. © 2010 Springer Science+Business Media, LLC.link_to_subscribed_fulltex

On the Optimal Stochastic Control of Dividend and Penalty Payments in an Insurance Company

In this thesis we consider the surplus of a non-life insurance company and assume that it follows either the classical Cramér-Lundberg model or its diffusion approximation. That is, we consider a continuous time model, where premiums are cashed at a constant rate and claims occur randomly with random sizes modelled by a compound Poisson process. In actuarial mathematics the risk of an insurance company is traditionally measured by the probability of ruin, where the time of ruin is defined as the first time when the surplus becomes negative. Using the ruin probability as a risk measure has been criticised because it is unrealistic to assume that an insurance company is ruined as soon as the surplus becomes negative. In this thesis, we assume that the insurer is not ruined although the surplus becomes negative. In order to avoid ruin, penalty payments occur, depending on the level of the surplus. For example, penalty payments occur if the insurance company needs to borrow money. In the first part of this thesis we consider the diffusion approximation to the Cramér-Lundberg model and we aim to determine a dividend strategy that maximises the difference between the expected discounted dividend and penalty payments, where penalty payments are either modelled by an exponential, linear or quadratic function. We show that the optimal strategy is a so-called barrier strategy and calculate the optimal barrier. The second part studies the analogous problem where the surplus process of an insurance company is given by a Cramér-Lundberg model. Here, similar results are obtained. In conclusion, we consider the problem where we have to determine an optimal investment and reinsurance strategy and the surplus follows the diffusion approximation. The insurance company can invest in several risky assets and reduce the insurance risk either by excess of loss or proportional reinsurance. The aim is to find a strategy which minimises the penalty payments that are necessary to avoid ruin. Various penalty functions are considered and closed form solutions are derived

Optimal stochastic control of dividends and capital injections

In this thesis, we consider optimisation problems of an insurance company whose risk reserve process follows the settings of the classical risk model. The insurer has the possibility to control its surplus process by paying dividends to the shareholders. Furthermore, the shareholders are allowed to make capital injections such that the surplus process stays nonnegative. A control strategy describes the decision on the times and the amount of the dividend payments and the capital injections. To measure the risk associated with a control strategy, we consider the value of the expected discounted dividends minus the penalised expected discounted capital injections. Because of the discounting, it can only be optimal to make capital injections at times when the reserve would become negative due to a claim occurrence. Our goal is to determine the value function, which is defined as the maximal value over all proper strategies, and to find an optimal strategy which leads to this maximal value. First, we solve the optimisation problem for the classical risk model when the capital injections are penalised by some proportional factor greater than one. We show that an optimal strategy exists and is of barrier type, i.e., all the surplus exceeding some barrier level is paid as dividend. In the second part, we extend this model by adding fixed costs incurring any time at which capital injections are made. The optimal strategy here is not of barrier type any more, but of band type. That is a strategy where the state space of the surplus process is partitioned into three types of sets where either dividends at the premium rate, or lump sum dividends, or no dividends at all are paid. In the third part, we allow the dynamics of the surplus process to depend on environmental conditions which are modelled by a Markov process. I.e., the frequency of the claim arrivals and the distribution of the claim amounts vary over time depending on the state of the environment process. We again maximise the difference between the expected discounted dividends and the (proportional) penalised capital injections and show that the optimal strategy for any fixed initial environment state is of barrier type with a barrier depending on the initial state