Multi-layer Dividend Strategy Research Articles

Recursive Approaches for Multi-Layer Dividend Strategies in a Phase-Type Renewal Risk Model

In this paper we consider a risk model with two independent classes of insurance risks in the presence of a multi-layer dividend strategy. We assume that both of the claim number processes are renewal processes with phase-type inter-arrival times. By analysing the Markov chains associated with the two given phase-type distributions of the inter-arrival times, algorithmic schemes for the determination of explicit expressions for the Gerber–Shiu expected discounted penalty function, as well as the expected discounted dividend payments are derived, using two different approaches.

On a Fractional Stochastic Risk Model with a Random Initial Surplus and a Multi-Layer Strategy

The paper deals with a fractional time-changed stochastic risk model, including stochastic premiums, dividends and also a stochastic initial surplus as a capital derived from a previous investment. The inverse of a ν-stable subordinator is used for the time-change. The submartingale property is assumed to guarantee the net-profit condition. The long-range dependence behavior is proven. The infinite-horizon ruin probability, a specialized version of the Gerber–Shiu function, is considered and investigated. In particular, we prove that the distribution function of the infinite-horizon ruin time satisfies an integral-differential equation. The case of the dividends paid according to a multi-layer dividend strategy is also considered.

Upper Bounds and Explicit Formulas for the Ruin Probability in the Risk Model with Stochastic Premiums and a Multi-Layer Dividend Strategy

This paper is devoted to the investigation of the ruin probability in the risk model with stochastic premiums where dividends are paid according to a multi-layer dividend strategy. We obtain an exponential bound for the ruin probability and investigate conditions, under which it holds for a number of distributions of the premium and claim sizes. Next, we use the exponential bound to construct non-exponential bounds for the ruin probability. We show that the non-exponential bounds turn out to be tighter than the exponential one in some cases. Moreover, we derive explicit formulas for the ruin probability when the premium and claim sizes have either the hyperexponential or the Erlang distributions and apply them to investigate how tight the bounds are. To illustrate and analyze the results obtained, we give numerical examples.

The risk model with stochastic premiums and a multi-layer dividend strategy

The paper deals with a generalization of the risk model with stochastic premiums where dividends are paid according to a multi-layer dividend strategy. First of all, we derive piecewise integro-differential equations for the Gerber–Shiu function and the expected discounted dividend payments until ruin. In addition, we concentrate on the detailed investigation of the model in the case of exponentially distributed claim and premium sizes and find explicit formulas for the ruin probability as well as for the expected discounted dividend payments. Lastly, numerical illustrations for some multi-layer dividend strategies are presented.

On the expected discounted penalty function for a risk model with dependence under a multi-layer dividend strategy

ABSTRACTIn this article, we consider a dependent risk model in the presence of a multi-laydividend strategy. We construct the dependence structure between the claim size and interclaim time by a Farlie–Gumbel–Morgenstern copula. A piecewise integro-differential equations for the expected discounted penalty function with boundary conditions are established. A renewal equation satisfied by the expected discounted penalty function is obtained via the translation operator. Then, we provide a recursive approach to derive the analytical solution of the expected discounted penalty function. Finally, a numerical example is presented to illustrate the solution procedure.

Markov-Dependent Risk Model with Multi-Layer Dividend Strategy and Investment Interest under Absolute Ruin

In this paper, we consider the Markov-dependent risk model with multi-layer dividend strategy and investment interest under absolute ruin, in which the claim occurrence and the claim amount are regulated by an external discrete time Markov chain. We derive systems of integro-differential equations satisfied by the moment-generating function, the nth moment of the discounted dividend payments prior to absolute ruin and the Gerber-Shiu function. Finally, the matrix form of systems of integro-differential equations satisfied by the Gerber-Shiu function is presented.

Markov-dependent risk model with multi-layer dividend strategy

In this paper, we propose a Markov-dependent risk model with multi-layer dividend strategy in which the claim occurrence and the claim amount are regulated by an external discrete time Markov chain. A system of piecewise integro-differential equations with boundary conditions satisfied by the Gerber–Shiu function, with given initial environment state, is derived. The closed form expression of the Gerber–Shiu function is obtained when all the claim amount distributions belong to the rational family. Also, we adopt the Chebyshev polynomial approach to find the approximate solution of the integro-differential equation. The numerical simulation results show the effectiveness of the proposed methods.

On a perturbed by diffusion compound Poisson risk model with delayed claims and multi-layer dividend strategy

In this paper we consider an extension to the compound Poisson risk process perturbed by diffusion in which two types of dependent claims, main claims and by-claims, are incorporated. Every by-claim is induced by the main claim and may be delayed for one time period with a certain probability. An integro-differential equation system for the Gerber–Shiu expected discounted penalty functions is derived and solved by proving that the Gerber–Shiu function satisfies some defective renewal equation. An exact representation for the solution of this equation is derived through an associated compound geometric distribution and an analytic expression for this quantity is given when both the claim and the by-claim amounts belong to the rational family of distributions. Further, the same risk model is considered in the presence of a multi-layer dividend strategy. A system of integro-differential equations for the expected discounted penalty functions depending on the current surplus level, with certain initial and boundary conditions, is obtained. To solve this, we derive a general solution to a certain second order integro-differential equation system. This solution is obtained by transforming this system to a Volterra-type system of integral equations of second kind, which is solved by using Laplace transforms provided an explicit expression for the Gerber–Shiu functions depending on the current surplus level. Finally, numerical results for the ruin probability are given to illustrate the applicability of our results.

The Gerber–Shiu discounted penalty function in a delayed renewal risk model with multi-layer dividend strategy

A class of delayed renewal risk processes with multi-layer dividend strategy is addressed here. Under the assumption that the premium rate is a step function depending on the current surplus level, a piecewise integro-differential equation for the Gerber–Shiu discounted penalty function in the delayed renewal risk model is derived, as an analogue of that in the ordinary renewal model, and the relationship between this function and the one in the ordinary renewal model is investigated. Subsequently, this relationship is detailed as regards both the stationary renewal risk model and the ruin probability. Finally, explicit expressions for some ruin-related quantities are included to illustrate the procedure, where the inter-claim times are generalized Erlang(2) distributed and the claims are exponentially distributed.

The discounted penalty function with multi-layer dividend strategy in the phase-type risk model

This paper considers a Sparre Andersen model in which the inter-claim times have a phase-type distribution and the premium rate is a step function depending on the current surplus level. We derive the system of piecewise integro-differential equations for the Gerber–Shiu discounted penalty functions and obtain the closed form expressions of the Gerber–Shiu functions if the claim amount distribution belongs to the rational family. We provide a recursive approach to calculate Gerber–Shiu functions and present an example.

The Discounted Penalty Function with Multi-Layer Dividend Strategy in the Phase-Type Risk Model

This paper considers a Sparre Andersen model in which the inter-claim times have a phase-type distribution and the premium rate is a step function depending on the current surplus level. We derive the system of piecewise integro-differential equations for the Gerber-Shiu discounted penalty functions and obtain the closed form expressions of the Gerber-Shiu functions if the claim amount distribution belongs to the rational family. We provide a recursive approach to calculate Gerber-Shiu functions and present an example.

On a Dual Model with Multi-Layer Dividend Strategy under Stochastic Interest

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 model of the compound Poisson model with multi-layer dividend strategy under stochastic interest. We derive a set of integro-differential equations satisfied by the expected total discounted dividends until ruin. The cases where profits follow an exponential distributions are solved.

On the Gerber–Shiu function for a risk model with multi-layer dividend strategy

In this paper, we present a new approach to the study of the Gerber–Shiu discounted function for the risk model with multi-layer dividend strategy. The formulae for the Gerber–Shiu discounted function and ruin probability were obtained and the special case where the claim size distribution is a combination of exponentials is considered in detail.

The compound Poisson risk model with dependence under a multi-layer dividend strategy

In this paper, a compound Poisson risk model with time-dependent claims is studied under a multi-layer dividend strategy. A piecewise integro-differential equation for the Gerber-Shiu function is derived and solved. Asymptotic formulas of the ruin probability are obtained when the claim size distributions are heavy-tailed.

On a perturbed Sparre Andersen risk model with multi-layer dividend strategy

In this paper, we consider a perturbed Sparre Andersen risk model, in which the inter-claim times are generalized Erlang(n) distributed. Under the multi-layer dividend strategy, piece-wise integro-differential equations for the discounted penalty functions are derived, and a recursive approach is applied to express the solutions. A numerical example to calculate the ruin probabilities is given to illustrate the solution procedure.

The perturbed compound Poisson risk model with multi-layer dividend strategy

In this paper, we consider a perturbed compound Poisson risk model with multi-layer dividend strategy. Integro-differential and integral equations for the expected discounted penalty function are derived and solved. When the claims are subexponentially distributed, the asymptotic formula for ruin probability is obtained.

Gerber–Shiu discounted penalty function in a Sparre Andersen model with multi-layer dividend strategy

This paper studies a Sparre Andersen model in which the inter-claim times are generalized Erlang(n) distributed. We assume that the premium rate is a step function depending on the current surplus level. A piecewise integro-differential equation for the Gerber–Shiu discounted penalty function is derived and solved. Finally, to illustrate the solution procedure, explicit expression for the Laplace transform of the time to ruin is given when the inter-claim times are generalized Erlang(2) distributed and the claim amounts are exponentially distributed.

Authors’ Reply: A Risk Model with Multilayer Dividend Strategy - Discussion by Cheung; Ramin Okhrati

Authors’ Reply: A Risk Model with Multilayer Dividend Strategy - Discussion by Cheung; Ramin Okhrati

“A Risk Model with Multilayer Dividend Strategy,” Hansjörg Albrecher and Jürgen Hartinger, April 2007

This is a discussion of a previously published paper, A Risk Model with Multilayer Dividend Strategy.

“A Risk Model with Multilayer Dividend Strategy”, Hansjorg Albrecher and Jürgen Hartinger, April 2007

“A Risk Model with Multilayer Dividend Strategy”, Hansjorg Albrecher and Jürgen Hartinger, April 2007