Dividend Barrier Research Articles

Optimal dividend policy with self-exciting claims in the Gamma–Omega model

In this paper, we consider the optimal dividend policy for an insurance company under a contagious insurance market, where the occurrence of a claim can trigger sequent claims. This clustering effect is modelled by a self-exciting Hawkes process where the intensity of claims depends on its historical path. In addition, we include the concept of bankruptcy to allow the insurance company to operate with a temporary negative surplus. The objective of the management is to obtain the optimal dividend strategy that maximises the expected discounted dividend payments until bankruptcy. The Hamilton–Jacobi–Bellman variational inequalities (HJBVIs) are derived rigorously. When claim sizes follow exponential distributions and the bankruptcy rate is a positive constant, the value function can be obtained based on the Gerber–Shiu penalty function and the optimal dividend barrier can be solved numerically. Finally, numerical examples are demonstrated to show the impact of key parameters on the optimal dividend strategy.

Perturbed MAP/PH Risk Model With Possible Delayed By-Claims and a Constant Dividend Barrier

Perturbed MAP/PH Risk Model With Possible Delayed By-Claims and a Constant Dividend Barrier

Gerber-Shiu theory for discrete risk processes in a regime switching environment

In this paper we develop the Gerber-Shiu theory for the classic and dual discrete risk processes in a Markovian (regime switching) environment. In particular, by expressing the Gerber-Shiu function in terms of potential measures of an upward (downward) skip-free discrete-time and discrete-space Markov Additive Process (MAP), we derive closed form expressions for the Gerber-Shiu function in terms of the so-called (discrete) Wv and Zv scale matrices, which were introduced in [27]. We show that the discrete scale matrices allow for a unified approach for identifying the Gerber-Shiu function as well as the value function of the associated constant dividend barrier problems.

Improving Risk Assessment and Pricing with Dividend Barriers and Dependence Modelling: An Extension of the Cramer-Lundberg Model with Spearman Copulas

Improving Risk Assessment and Pricing with Dividend Barriers and Dependence Modelling: An Extension of the Cramer-Lundberg Model with Spearman Copulas

On the dual risk model with Parisian implementation delays under a mixed dividend strategy

AbstractIn this paper, we consider a mixed dividend strategy in a dual risk model. The mixed dividend strategy is the combination of a threshold dividend and a Parisian implementation delays dividend under periodic observation. Given a series of discrete observation points, when the surplus level is larger than the predetermined bonus barrier at observation point, the Parisian implementation delays dividend is immediately carried out, and the threshold dividend is performed continuously during the delayed period. We study the Gerber-Shiu expected discounted penalty function and the expected discounted dividend payments before ruin in such a dual risk model. Numerical illustrations are given to study the influence of relevant parameters on the ruin-related quantities and the selection of the optimal dividend barrier for a given initial surplus level.

A Markovian risk model with possible by-claims and dividend barrier

A Markovian risk model with possible by-claims and dividend barrier

The Risk Model with a Constant Dividend Barrier Affected by a Threshold Value

This paper considers a new risk model with a constant dividend barrier, which the claim amount affected by a threshold value. The hypothesis of the model is presented and the integro-differential equation for the Gerber-Shiu penalty function is given. Then the linear solution of the Gerber-Shiu discounted penalty function is figured out. The paper also derives the integro-differential equation and the linear solution of the expected discounted dividend payments. An example is given too.

Moments of deficit duration and its proportion in general compound binomial model

This paper discusses the deficit duration in the general compound binomial model. We derive the recursive formulas of moments of the deficit duration and its proportion in finite time with and without a dividend barrier, and use operators to design algorithms for finding their solutions. Under the condition of positive safety loading, we discuss the limit of deficit duration and show the convergence of moments of its proportion. In addition, the paper also discusses a sort of reinsurance strategy to significantly reduce the deficit duration. The proposed algorithms and reinsurance strategy are effective and are illustrated by numerical examples.

Infinite series expansion of some finite-time dividend and ruin related functions

In this paper, we consider some finite-time dividend and ruin problems in a class of risk models under the constant dividend barrier strategy. This class of risk models includes Brownian motion risk model and the compound Poisson model with and without diffusion. By using Laguerre series expansion, we derive infinite series expansion formulas for the time T-deferred Laplace transform of the ruin time and time T-deferred expected present valued of dividend payments until ruin.

On Itô’s formula for semimartingales with jumps and non-[formula omitted]2 functions

This paper considers a variant of Itô’s formula for discontinuous semimartingales and non-C2 functions. This result is particularly helpful for insurance control problems with Markov-modulated components. An example of a dividend barrier strategy for a Brownian risk model with Markov-switching illustrates the result.

A Markovian risk model with possible by-claims and dividend barrier

A Markovian risk model with possible by-claims and dividend barrier

Gerber-Shiu Function in a Discrete-time Risk Model with Dividend Strategy

In this paper, a discrete-time risk model with dividend strategy and a general premium rate is considered. Under such a strategy, once the insurer’s surplus hits a constant dividend barrier , dividends are paid off to shareholders at instantly. Using the roots of a generalization of Lundberg’s fundamental equation and the general theory on difference equations, two difference equations for the Gerber-Shiu discounted penalty function are derived and solved. The analytic results obtained are utilized to derive the probability of ultimate ruin when the claim sizes is a mixture of two geometric distributions. Numerical examples are also given to illustrate the applicability of the results obtained.

Ruin and Dividend Measures in the Renewal Dual Risk Model

In this manuscript we consider the dual risk model with financial application, where the random gains occur under a renewal process. We particularly work the Erlang(n) case for common distribution of the inter-arrival times, from there it is easy to understand that our method or procedures can be generalised to other cases under the matrix-exponential family case. We work several and different problems involving future dividends and ruin. We also show that our results are valid even if the usual income condition is not satisfied. In most known works under the dual model, the main target under study have been the calculation of expected discounted future dividends and optimal strategies, where the dividend calculation have been done on aggregate. We can find works, at first using the classical compound Poisson model, then some examples of other renewal Erlang models. Knowing that ruin is ultimately achieved, we find important that dividends should be evaluated on an individual basis, where the early dividend contribution for the aggregate are of utmost importance. From our calculations we can really see how much important is the contribution of the first dividend. Afonso et al. (Insur Math Econ, 53(3), 906–918, 2013) had worked similar problems for the classical compound Poisson dual model. Besides that we find explicit formulae for both the probability of getting a dividend and the distribution of the amount of a single dividend. We still work the probability distribution of the number of gains to reach a given upper target (like a constant dividend barrier) as well as for the number of gains down to ruin. We complete the study working some illustrative numerical examples that show final numbers for the several problems under study.

On a Discrete-Time Risk Model with Random Income and a Constant Dividend Barrier

In this paper, a discrete-time risk model with random income and a constant dividend barrier is considered. Under such a dividend policy, once the insurer’s reserve hits the level b b > 0 , the excess of the reserve over b is paid off as dividends. We derive a homogeneous difference equation for the expected present value of dividend payments. Corresponding solution procedures for the difference equation are invested. Finally, we give a numerical example to illustrate the applicability of the results obtained.

Randomized observation periods for compound Poisson risk model with capital injection and barrier dividend

In this paper, we model the insurance company’s surplus by a compound Poisson risk model, where the surplus process can only be observed at random observation times. It is assumed that the insurer observes its surplus level periodically to decide on dividend payments and capital injection at the interobservation time having an operatorname{Erlang}(n) distribution. If the observed surplus level is greater than zero but less than injection line b_{1} > 0, the shareholders should immediately inject a certain amount of capital to bring the surplus level back to the injection line b_{1}. If the observed surplus level is larger than dividend line b_{2} (b_{2} > b_{1}), any excess of the surplus over b_{2} is immediately paid out as dividends to the shareholders of the company. Ruin is declared when the observed surplus level is negative. We derive the explicit expressions of the Gerber–Shiu function, the expected discounted capital injection, and the expected discounted dividend payments. Numerical illustrations are also given to analyze the effect of random observation times on actuarial quantities.

The Laplace transform of ruin time with investment and barrier dividend

In order to study the Gerber-Shiu function for a compound Poisson-Geometric risk model with investment and barrier dividend, we obtain the differential-integral equation of the Gerber-Shiu function by using the method of total expectation formula. Under the assumption of exponential distribution, the explicit solution of the Laplace transform of ruin time of insurance company with investment and barrier dividend is given.

Optimal Impulse Control for Growth-Restricted Linear Diffusions with Regime Switching

This paper investigates the impulse cash-flow withdrawal (dividend distribution or consumption) optimization problem for a broad class of growth restricted linear diffusions with drift and volatility dependent on both the level of surplus and the environment/economy regime (described by a Markov chain). There are proportional and fixed costs associated with each cash-flow withdrawal transaction, and the objective of the optimization problem is to determine the optimal policy that maximizes the expected total cumulative discounted cash-flow withdrawals (dividend payments/consumption) minus expenses. This work extends all the existing work on the same problem under the diffusion setting by extending the underlying dynamics to a much more general class of Markov modulated diffusions and unifies the existing work on optimal impulse dividend control. We show that when there are two regimes, the value function, as a function of the initial cash reservoir and the initial environmental state, can be determined by solving a set of second order ordinary differential equations. We further show that either the optimal strategy is a regime switching lump-sum dividend barrier strategy or there exists no optimal strategies.

On the improved thinning risk model under a periodic dividend barrier strategy

<abstract><p>In this study, we consider a periodic dividend barrier strategy in an improved thinning risk model, which indicates that insurance companies randomly receive premiums and pay dividends. In the improved model, the premium is stochastic, and the claim counting process is a p-thinning process of the premium counting process. The integral equations satisfied by the Gerber-Shiu function and the expected discounted cumulative dividend function are derived. Explicit expressions of those actuarial functions are obtained when the claim and premium sizes are exponentially distributed. We analyze and illustrate the impact of various parameters on them and obtain the optimal barrier. Finally, a conclusion is drawn.</p></abstract>

Draw-down Parisian ruin for spectrally negative Lévy processes

AbstractDraw-down time for a stochastic process is the first passage time of a draw-down level that depends on the previous maximum of the process. In this paper we study the draw-down-related Parisian ruin problem for spectrally negative Lévy risk processes. Intuitively, a draw-down Parisian ruin occurs when the surplus process has continuously stayed below the dynamic draw-down level for a fixed amount of time. We introduce the draw-down Parisian ruin time and solve the corresponding two-sided exit problems via excursion theory. We also find an expression for the potential measure for the process killed at the draw-down Parisian time. As applications, we obtain new results for spectrally negative Lévy risk processes with dividend barrier and with Parisian ruin.

Optimal dividend and risk control policies in the presence of a fixed transaction cost

In this paper, we consider a large insurance company whose cumulative cash flow process is described by a drifted Brownian motion. The preference of the insurer is to maximize his/her firm’s value, which corresponds to the expected present value of the dividend payments up to the ruin time. In the business process, the insurer has the option to draw up a dividend payment policy and to purchase proportional reinsurance at a certain point in time. In view of some typical practical expenses (e.g., consultant commission), it is assumed that a fixed transaction cost occurs at the beginning of the reinsurance commitment which, once made, is irreversible. This leads to a mixed stochastic control problem of optimal stopping time and singular control. For this mixed problem, we derive closed-form solutions for the optimal time to purchase reinsurance, the optimal retained risk proportion, the optimal dividend barrier, and the value function. The optimal solution shows that reinsurance is valueless to the insurer when the fixed cost is larger than a threshold, and comes into play when the fixed cost is less than this threshold. We also perform some numerical calculations to assess the impacts of fixed costs on the value function and the optimal policies.