A Markov decision problem in a risk model with interest rate and Markovian environment

We consider the compound binomial model in a Markovian environment presented by Cossette et al.(2004). We modify the model via assuming that the company receives interest on the surplus and a positive real-valued premium per unit time, and introducing a control strategy of periodic dividend payments. A Markov decision problem arises and the control objective is to maximize the cumulative expected discounted dividends paid to the shareholders until ruin minus a discounted penalty for ruin. We show that under the absence of a ceiling of dividend rates the optimal strategy is a conditional band strategy given the current state of the environment process. Under the presence of a ceiling for dividend rates, the character of the optimal control strategy is given. In addition, we offer an algorithm for the optimal strategy and the optimal value function.Numerical results are provided to illustrate the algorithm and the impact of the penalty.     

On the Gerber-Shiu Function and Optimal Dividend Strategy for a Thinning Risk Model

In this paper, the risk model under constant dividend barrier strategy is studied, in which the premium income follows a compound Poisson process and the arrival of the claims is a p-thinning process of the premium arrival process. The integral equations with boundary conditions for the expected discounted aggregate dividend payments and the expected discounted penalty function until ruin are derived. In addition, the explicit expressions for the Laplace transform of the ruin time and the expected aggregate discounted dividend payments until ruin are given when the individual stochastic premium amount and claim amount are exponentially distributed. Finally, the optimal barrier is presented under the condition of maximizing the expectation of the difference between discounted aggregate dividends until ruin and the deficit at ruin.     

Optimal dividends in the dual model

The optimal dividend problem proposed by de Finetti [de Finetti, B., 1957. Su un’impostazione alternativa della teoria collettiva del rischio. In: Transactions of the XVth International Congress of Actuaries, vol. 2. pp. 433-443] is to find the dividend-payment strategy that maximizes the expected discounted value of dividends which are paid to the shareholders until the company is ruined or bankrupt. In this paper, it is assumed that the surplus or shareholders’ equity is a Lévy process which is skip-free downwards; such a model might be appropriate for a company that specializes in inventions and discoveries. In this model, the optimal strategy is a barrier strategy. Hence the problem is to determine b^∗, the optimal level of the dividend barrier. A key tool is the method of Laplace transforms. A variety of numerical examples are provided. It is also shown that if the initial surplus is b^∗, the expectation of the discounted dividends until ruin is the present value of a perpetuity with the payment rate being the drift of the surplus process.     

Obtaining the dividends-penalty identities by interpretation

The dividends-penalty identity is a relation between three functions: the discounted penalty function without dividends, the discounted penalty function if a barrier dividend strategy is applied, and the expected discounted dividends until ruin. The classical model of risk theory is modified in that the deterministic premiums are replaced by a compound Poisson process with exponential jumps. In this model, the dividends-penalty identity is new and can be derived by interpretation. Then the dividends-penalty identity in the classical model is obtained as a limit.     

Dividend payments with a threshold strategy in the compound Poisson risk model perturbed by diffusion

In the absence of dividends, the surplus of an insurance company is modelled by a compound Poisson process perturbed by diffusion. Dividends are paid at a constant rate whenever the modified surplus is above the threshold, otherwise no dividends are paid. Two integro-differential equations for the expected discounted dividend payments prior to ruin are derived and closed-form solutions are given. Accordingly, the Gerber–Shiu expected discounted penalty function and some ruin related functionals, the probability of ultimate ruin, the time of ruin and the surplus before ruin and the deficit at ruin, are considered and their analytic expressions are given by general solution formulas. Finally the moment-generating function of the total discounted dividends until ruin is discussed.     

Dividend problems in the dual model with diffusion and exponentially distributed observation time

Consider dividend problems in the dual model with diffusion and exponentially distributed observation time where dividends are paid according to a barrier strategy. Assume that dividends can only be paid with a certain probability at each point of time, that is, on each observation, if the surplus exceeds the barrier, the excess is paid as dividend. In this paper, integro-differential equations for the expected discounted sum of dividends paid until ruin and the Laplace transform of ruin time are derived. When the gains are exponentially distributed, explicit expressions for the ruin probability, the expected discounted sum of dividends paid until ruin, the Laplace transform of ruin time and the expectation of ruin time are also obtained.     

The Markov Additive Risk Process Under an Erlangized Dividend Barrier Strategy

In this paper, we consider a Markov additive insurance risk process under a randomized dividend strategy in the spirit of Albrecher et al. (2011). Decisions on whether to pay dividends are only made at a sequence of dividend decision time points whose intervals are Erlang(n) distributed. At a dividend decision time, if the surplus level is larger than a predetermined dividend barrier, then the excess is paid as a dividend as long as ruin has not occurred. In contrast to Albrecher et al. (2011), it is assumed that the event of ruin is monitored continuously (Avanzi et al. (2013) and Zhang (2014)), i.e. the surplus process is stopped immediately once it drops below zero. The quantities of our interest include the Gerber-Shiu expected discounted penalty function and the expected present value of dividends paid until ruin. Solutions are derived with the use of Markov renewal equations. Numerical examples are given, and the optimal dividend barrier is identified in some cases.     

The Risk Model with Interest,Liquid Reserves and a Constant Dividend Barrier

In this paper, we consider the compound Poisson surplus model with interest, liquid reserves and a constant dividend barrier. When the surplus of an insurer is below a fixed level, the surplus is kept as liquid reserves, which does not earn interest. When the surplus attains the level, the surplus will receive interest at a constant rate. When the surplus hits another fixed higher lever, the excess of the surplus over this higher level will be distributed to the shareholders as dividends. We derive a system of integro-differential equations for the Gerber-Shiu discounted penalty function and obtain the solutions to these integro-differential equations. In the case where the claim sizes are exponential distributed, we get the exact solutions of zero discounted Gerber-Shiu function. We also get the integro-differential equation for the expectation of the discounted dividends until ruin which is the key to discuss the optimal dividend barrier. And we give the exact solution in the special case with exponential claim sizes.     

Dividend optimization under the gamma-distribution of claims

The classic insurance company work model with gamma-distribution of claim amount is considered. It is supposed that the company applies a dividend barrier strategy. The form of the expected discounted dividends accumulated until the ruin and the expected discounted deficit at the ruin are found. We deal with the optimal barriers which maximize either the dividends amount or shareholders profit. The barrier optimization is illustrated by some examples.     

Lévy risk model with two-sided jumps and a barrier dividend strategy

In this paper, we consider a general Lévy risk model with two-sided jumps and a constant dividend barrier. We connect the ruin problem of the ex-dividend risk process with the first passage problem of the Lévy process reflected at its running maximum. We prove that if the positive jumps of the risk model form a compound Poisson process and the remaining part is a spectrally negative Lévy process with unbounded variation, the Laplace transform (as a function of the initial surplus) of the upward entrance time of the reflected (at the running infimum) Lévy process exhibits the smooth pasting property at the reflecting barrier. When the surplus process is described by a double exponential jump diffusion in the absence of dividend payment, we derive some explicit expressions for the Laplace transform of the ruin time, the distribution of the deficit at ruin, and the total expected discounted dividends. Numerical experiments concerning the optimal barrier strategy are performed and new empirical findings are presented.     

Dividend maximization under consideration of the time value of ruin

In the Cramér-Lundberg model and its diffusion approximation, it is a classical problem to find the optimal dividend payment strategy that maximizes the expected value of the discounted dividend payments until ruin. One often raised disadvantage of this approach is the fact that such a strategy does not take the lifetime of the controlled process into account. In this paper we introduce a value function which considers both expected dividends and the time value of ruin. For both the diffusion model and the Cramér-Lundberg model with exponential claim sizes, the problem is solved and in either case the optimal strategy is identified, which for unbounded dividend intensity is a barrier strategy and for bounded dividend intensity is of threshold type.     

Parisian ruin with a threshold dividend strategy under the dual Lévy risk model

We consider the threshold dividend strategy where a company’s surplus process is described by the dual Lévy risk model. Namely, the company chooses to pay dividends at a constant rate only when the surplus is above some nonnegative threshold. Classically, such a company is referred to be ruined immediately when the surplus level becomes negative. Recently, researchers investigate the Parisian ruin problem where the company is allowed to operate under negative surplus for a predetermined period known as the Parisian delay. With the help of the fluctuation identities of spectrally negative Lévy processes, we obtain an explicit expression of the expected discounted dividends until Parisian ruin in terms of the relevant scale functions and certain probabilities that need to be evaluated for each specific Lévy process. The optimal threshold level under such a threshold dividend strategy is deduced. Applications and numerical examples are given to illustrate the theoretical results and examine how the expected discounted aggregate dividends and the optimal threshold level change in response to different Parisian delays.     

Optimality of the threshold dividend strategy for the compound Poisson model

In this paper, we consider the optimal dividend problem for the compound Poisson risk model. We assume that dividends are paid to the shareholders according to an admissible strategy with dividend rate bounded by a constant. Our objective is to find a dividend policy so as to maximize the expected discounted value of dividends until ruin. We give sufficient conditions under which the optimal strategy is of threshold type.     

Optimal control problem for an insurance surplus model with debt liability

For an insurance company with a debt liability, they could make some management actions, such as reinsurance, paying dividends, and capital injection, to balance the profitability and financial bankruptcy. Our objective is to determine risk retention rate, dividend, and capital injection strategy so as to maximize the expected discounted dividends minus the discounted cost of capital injection until the time of ruin. We assume that the dividend payments and capital injection should occur with both fixed and proportional costs. We obtain explicit expressions of the optimal value functions as well as the corresponding optimal joint strategies by routine procedures in a comprehensive basic model using a new technique to solve the related equations. Our results show that whether recapitalizing is profitable or not depends on the costs of capital raising and that the firm injects capital only when the reserves are zero and recapitalizes to the optimal reserves level if the cost of external capital is low. Copyright © 2013 John Wiley & Sons, Ltd.     

Optimal dividends in the Brownian motion risk model with interest

In this paper, we consider a Brownian motion risk model, and in addition, the surplus earns investment income at a constant force of interest. The objective is to find a dividend policy so as to maximize the expected discounted value of dividend payments. It is well known that optimality is achieved by using a barrier strategy for unrestricted dividend rate. However, ultimate ruin of the company is certain if a barrier strategy is applied. In many circumstances this is not desirable. This consideration leads us to impose a restriction on the dividend stream. We assume that dividends are paid to the shareholders according to admissible strategies whose dividend rate is bounded by a constant. Under this additional constraint, we show that the optimal dividend strategy is formed by a threshold strategy.     

Dividends in finite time horizon

In this paper, we consider the classical risk model modified in two different ways by the inclusion of a dividend barrier. For Model I, we present numerical algorithms, which can be used to approximate or bound the expected discounted value of dividends up to a finite time horizon, t, or ruin if this occurs earlier. We extend this by requiring the shareholders to provide the initial capital and to pay the deficit at ruin each time it occurs so that the process then continues after ruin up to time t. For Model I, we assume the full premium income is paid as dividends whenever the surplus exceeds a set level. In our Model II, we assume dividends are paid at a rate less than the rate of premium income. Copyright © 2012 John Wiley & Sons, Ltd.     

Optimal dividend strategies in discrete risk model with capital injections

In this paper we consider a doubly discrete model used in Dickson and Waters (biASTIN Bulletin 1991; 21 :199–221) to approximate the Cramér–Lundberg model. The company controls the amount of dividends paid out to the shareholders as well as the capital injections which make the company never ruin in order to maximize the cumulative expected discounted dividends minus the penalized discounted capital injections. We show that the optimal value function is the unique solution of a discrete Hamilton–Jacobi–Bellman equation by contraction mapping principle. Moreover, with capital injection, we reduce the optimal dividend strategy from band strategy in the discrete classical risk model without external capital injection into barrier strategy , which is consistent with the result in continuous time. We also give the equivalent condition when the optimal dividend barrier is equal to 0. Although there is no explicit solution to the value function and the optimal dividend barrier, we obtain the optimal dividend barrier and the approximating solution of the value function by Bellman's recursive algorithm. From the numerical calculations, we obtain some relevant economical insights. Copyright © 2010 John Wiley & Sons, Ltd.     

On optimal periodic dividend strategies in the dual model with diffusion

The dual model with diffusion is appropriate for companies with continuous expenses that are offset by stochastic and irregular gains. Examples include research-based or commission-based companies. In this context, Bayraktar et al. (2013a) show that a dividend barrier strategy is optimal when dividend decisions are made continuously. In practice, however, companies that are capable of issuing dividends make dividend decisions on a periodic (rather than continuous) basis.In this paper, we consider a periodic dividend strategy with exponential inter-dividend-decision times and continuous monitoring of solvency. Assuming hyperexponential gains, we show that a periodic barrier dividend strategy is the periodic strategy that maximizes the expected present value of dividends paid until ruin. Interestingly, a ‘liquidation-at-first-opportunity’ strategy is optimal in some cases where the surplus process has a positive drift. Results are illustrated.     

On a dual model with a dividend threshold

In insurance mathematics, a compound Poisson model is often used to describe the aggregate claims of the surplus process. In this paper, we consider the dual of the compound Poisson model under a threshold dividend strategy. We derive a set of two integro-differential equations satisfied by the expected total discounted dividends until ruin and show how the equations can be solved by using only one of the two integro-differential equations. The cases where profits follow an exponential or a mixture of exponential distributions are then solved and the discussion for the case of a general profit distribution follows by the use of Laplace transforms. We illustrate how the optimal threshold level that maximizes the expected total discounted dividends until ruin can be obtained, and finally we generalize the results to the case where the surplus process is a more general skip-free downwards Lévy process.     

Discounted dividends in a strategy with a step barrier function

A model of insurance company performance with dividend payment is studied. We investigate the dividend strategy according to which the level of the barrier can be changed after the receipt of claims. A function representing the value of expected discounted dividends paid until ruin is obtained.