Threshold Dividend Strategy Research Articles

Numerical method for a compound Poisson risk model with liquid reserves and proportional investment

<abstract><p>In this paper, a classical risk model with liquid reserves and proportional investment is considered, and the expected total discounted dividend before ruin of insurance companies under the threshold dividend strategy is studied. First, the integral differential equations of the expected total discounted dividend before ruin satisfying certain boundary conditions is derived. Second, since the explicit solutions of the equations cannot be obtained, the numerical approximation solutions are obtained by the sinc approximation method. Finally, we discuss the effects of parameters such as risk capital ratio and liquid reserve on the expected total discounted dividend before ruin by some examples.</p></abstract>

On the Expected Discounted Penalty Function Using Physics-Informed Neural Network

We study the expected discounted penalty at ruin under a stochastic discount rate for the compound Poisson risk model with a threshold dividend strategy. The discount rate is modeled by a Poisson process and a standard Brownian motion. By applying the differentiation method and total expectation formula, we obtain an integrodifferential equation for the expected discounted penalty function. From this integrodifferential equation, a renewal equation and an asymptotic formula satisfied by the expected discounted penalty function are derived. In order to solve the integrodifferential equation, we use a physics-informed neural network (PINN) for the first time in risk theory and obtain the numerical solutions of the expected discounted penalty function in some special cases of the penalty at ruin.

A refracted Lévy process with delayed dividend pullbacks

ABSTRACT The threshold dividend strategy, under which dividends are paid only when the insurer's surplus exceeds a pre-determined threshold, has received considerable attention in risk theory. However, in practice, it seems rather unlikely that an insurer will immediately pull back the dividend payments as soon as its surplus level drops below the dividend threshold. Hence, in this paper, we propose a refracted Lévy risk model with delayed dividend pullbacks triggered by a certain Poissonian observation scheme. Leveraging the extensive literature on fluctuation identities for spectrally negative Lévy processes, we obtain explicit expressions for two-sided exit identities of the proposed insurance risk process. Also, penalties are incorporated into the analysis of dividend payouts as a mechanism to penalize for the volatility of the dividend policy and account for an investor's typical preference for more stable cash flows. An explicit expression for the expected (discounted) dividend payouts net of penalties is derived. The criterion for the optimal threshold level that maximizes the expected dividend payouts is also discussed. Finally, several numerical examples are considered to assess the impact of dividend delays on ruin-related quantities. We numerically show that dividend strategies with more steady dividend payouts can be preferred (over the well-known threshold dividend strategy) when penalty fee become too onerous.

Numerical Method for a Perturbed Risk Model with Proportional Investment

In this paper, we study the perturbed risk model with a threshold dividend strategy and proportional investment. The insurance companies are allowed to invest their surplus in a financial market consisting of a risk-free asset and a risky asset in fixed proportions; the risky assets are modeled by the jump-diffusion process. Firstly, using the theory of the stochastic process and stochastic analysis, we obtained the integro-differential equations satisfied by the expected discounted dividend payments and the discounted penalty function. Secondly, we obtained the numerical approximate solutions of the integro-differential equations through the sinc method, since the analytical solutions of them are not easy to obtain, and we found that the error is within a manageable range. Finally, we considered some numerical examples where the claim sizes follow an exponential distribution, a mixture of two exponential distributions or the lognormal distribution in detail, and explored how perturbations and proportional investment affect dividends and ruin probability. Moreover, sensitive analysis showed that the proportion of the risky investment, the diffusion coefficient, the distribution of the claims and the positive jump in the risky assets investment all have explicit impacts on dividends and ruin probability.

A markov-modulated risk model with transaction costs and threshold dividend strategy

This paper concerns the numerical solution of the expected discounted dividend payments in a Markov-modulated risk model with investment, transaction costs and threshold dividend strategy. A system of delay integro-differential equations with certain boundary conditions is derived. As a closed-form solution does not exist, a numerical sinc method is proposed. Furthermore, an example is considered to illustrate the effect of the transaction costs, investment strategy and market environment.

Omega Model for a Jump-Diffusion Process with a Two-Step Premium Rate and a Threshold Dividend Strategy

In this paper, a jump-diffusion Omega model with a two-step premium rate and a threshold dividend strategy is studied. For this model, the surplus process is a perturbation of a compound Poisson process by a Brownian motion. Firstly, using the strong Markov property, the integro-differential equations for the expected discounted dividend payments function, the Gerber-Shiu expected discounted penalty function and bankruptcy probability are derived. Secondly, for a constant bankruptcy rate function, the renewal equations satisfied by the expected discounted dividend payments function and the Gerber-Shiu expected discounted penalty function are obtained, respectively, and by iteration, their closed-form solutions are also given. Furthermore, the explicit solutions of the two kinds of functions are obtained when the individual claim size is subject to exponential distribution. Finally, a numerical example is presented to illustrate some properties of the Omega model.

On Optimality of Threshold Dividend Strategy for Regime-Switching Diffusions With Negative Exponential Jumps

On Optimality of Threshold Dividend Strategy for Regime-Switching Diffusions With Negative Exponential Jumps

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.

Dividends under threshold dividend strategy with randomized observation periods and capital-exchange agreement

This paper considers the Cramér–Lundberg risk model for two companies with capital-exchange agreement, threshold dividend strategies and randomized observation periods. A integro-partial differential system of equations that characterizes the expected discounted dividend payments is obtained. As closed-form solutions do not exist, a numerical Sinc-Collocation method is proposed. Finally, an example illustrating the procedure is presented.

The risk model with stochastic premiums, dependence and a threshold dividend strategy

The paper deals with a generalization of the risk model with stochastic premiums where dependence structures between claim sizes and inter-claim times as well as premium sizes and inter-premium times are modeled by Farlie--Gumbel--Morgenstern copulas. In addition, dividends are paid to its shareholders according to a threshold dividend strategy. We derive integral and integro-differential equations for the Gerber--Shiu function and the expected discounted dividend payments until ruin. Next, we concentrate on the detailed investigation of the model in the case of exponentially distributed claim and premium sizes. In particular, we find explicit formulas for the ruin probability in the model without either dividend payments or dependence as well as for the expected discounted dividend payments in the model without dependence. Finally, numerical illustrations are presented.

The compound Poisson risk model under a mixed dividend strategy

In this paper, we consider a compound Poisson model under a mixed dividend strategy. The mixed dividend strategy is a combination of threshold dividend strategy and periodic dividend strategy. Given a positive threshold level b > 0, whenever the surplus process attains the level b, dividends will be paid off continuously at a rate α > 0. Furthermore, given a sequence of dividend decision times {Zj}j=1∞, whenever the observed surplus level at Zj is larger than b, the excess value will also be paid off as dividend. We study the expected discounted dividend payments before ruin and the Gerber–Shiu expected discounted penalty function. Some numerical examples are also presented.

二维对偶模型的最优分红

研究阈值分红策略下带扩散的二维对偶模型，得到分红现值的期望所满足的积分微分方程组，并用此方程组解得收入服从指数分布时的分红折现的期望具体表达式，应用拉普拉斯变换求得收益服从任意分布时的解，最后解得了此模型下的最优边界。 We study the two-dimensional dual model with diffusion under a threshold dividend strategy. We obtain a group of integro-differential equations satisfied by the expected total sum of discount dividends until ruin. And explicit results when the gains of the two projects are exponentially distributed are derived. By applying the method of the Laplace transform, we solve the case where gains follow general distributions. We also illustrate how to calculate the optimal dividend threshold.

Closed-Form Absolute Ruin Problems of the Risk Models with State-Dependent Switched Claims

This letter mainly investigates a general risk model with the threshold dividend strategy under assumption that the claim amounts obey a state-dependent switched exponential distribution. By establishing the differential-integral equations for the Gerber-Shiu discounted penalty function, and applying the hypergeometric functions, the closed-form absolute ruin probability is derived.

On the joint analysis of the total discounted payments to policyholders and shareholders: threshold dividend strategy

AbstractIn insurance risk theory, dividend and aggregate claim amount are of great research interest as they represent the insurance company’s payments to its shareholders and policyholders, respectively. Since the analyses of these two quantities are performed separately in the literature, the companion paper by Cheung et al. generalised the Gerber–Shiu expected discounted penalty function by further incorporating the moments of the aggregate discounted claims until ruin and the discounted dividends until ruin. While Cheung et al. considered the compound Poisson model with a dividend barrier in which ruin occurs almost surely, the present paper looks at this generalised Gerber–Shiu function under a threshold dividend strategy where the insurer has a positive survival probability. Because the Gerber–Shiu function is only defined for sample paths leading to ruin, we will additionally study the joint moments of the aggregate discounted claims and the discounted dividends without ruin occurring. Some explicit formulas are derived when the individual claim distribution follows a combination of exponentials. Numerical illustrations involving the correlation between aggregate discounted claims and discounted dividends are given. For the case where ruin occurs, we additionally compute the correlations between the time of ruin and the above two quantities.

On a dual risk model perturbed by diffusion with dividend threshold

In the dual risk model, the surplus process of a company is a Levy process with sample paths that are skip-free downwards. In this paper, the authors assume that the surplus process is the sum of a compound Poisson process and an independent Wiener process. The dual of the jump-diffusion risk model under a threshold dividend strategy is discussed. The authors derive a set of two integro-differential equations satisfied by the expected total discounted dividend until ruin. The cases where profits follow an exponential or mixtures of exponential distributions are solved. Applying the key method of the Laplace transform, the authors show how the integro-differential equations are solved. The authors also discuss the conditions for optimality and show how an optimal dividend threshold can be calculated as well.

Markov-modulated, multi-threshold dual risk model

In this paper we consider a Markov-modulated dual risk reserve process with a multi–threshold dividend strategy. We study the risk reserve process that can start at any level of the threshold. An integral equation for the conditional non–ruin probability is obtained. Further in two states, we consider the process influenced by two models. In the first model the process can be in two different states and the state can change at the arrival of a random gain with respect to the threshold barrier. In the second model the state changes at the arrival of either a random gain or an independent Poisson observer. The explicit form of the survival probabilities for both the states irrespective of the initial state being 1 or 2 and the process starting below or above the barrier are obtained when the random gain size distribution is exponential. Finally, numerical examples illustrate the main results.

A Markov-modulated jump-diffusion risk model with randomized observation periods and threshold dividend strategy

This paper considers a Markov-modulated jump-diffusion risk model with randomized observation periods and threshold dividend. A second order integro-differential system of equations that characterizes the expected discounted dividend payments is obtained. As a closed-form solution does not exist, a numerical procedure based on the sinc function approximation through a collocation method is proposed. Finally, an example illustrating the procedure is presented.

The ruin time under the Sparre-Andersen dual model

In this paper, we study the Sparre-Andersen dual risk model in which the times between positive gains are independently and identically distributed and have a generalized Erlang-n distribution. An important difference between this model and some other models such as the Erlang-n dual risk model is that the roots to the generalized Lundberg’s equation are not necessarily distinct. Hence, we derive an explicit expression for the Laplace transform of the ruin time, which involves multiple roots. Also, we apply our approach for obtaining the expected discounted dividends when the threshold-dividend strategy discussed by Ng (2009) is implemented under the Sparre-Andersen model with Erlang-n distribution of the inter-event times. In particular, we derive an explicit form of the expected discounted dividends when jump sizes are exponential.

On the compound Poisson risk model with dependence and a threshold dividend strategy

In this paper, we consider the compound Poisson risk model with a threshold dividend strategy and a dependence structure modeled by a Farlie–Gumbel–Morgenstern copula. The integro-differential equations satisfied by the Gerber–Shiu functions and the expected discounted dividend payments paid until ruin respectively are derived. Further, by deriving and solving the renewal equations satisfied by the Gerber–Shiu functions and the expected discounted dividend payments, we give the explicit formulas for them.

On a perturbed Sparre Andersen risk model with threshold dividend strategy and dependence

In this paper, we consider a Sparre Andersen risk model perturbed by a Brownian motion, where the individual claim sizes are dependent on the interclaim times. We assume that dividends are paid off under a threshold strategy. Integral and integro-differential equations satisfied by the Gerber–Shiu functions are obtained, and a solution procedure is also proposed.