On a perturbed MAP risk model under a threshold dividend strategy

In this paper, we consider a perturbed risk model where the claims arrive according to a Markovian arrival process (MAP) under a threshold dividend strategy. We derive the integro-differential equations for the Gerber–Shiu expected discounted penalty function and the moments of total dividend payments until ruin, obtain the analytical solutions to these equations, and give numerical examples to illustrate our main results. We also get a matrix renewal equation for the Gerber–Shiu function, and present some asymptotic formulas for the Gerber–Shiu function when the claim size distributions are heavy-tailed.     

On the Gerber–Shiu discounted penalty function in a risk model with delayed claims

In this paper, we consider an extension to the continuous time risk model for which the occurrence of the claim may be delayed and the time of delay for the claim is assumed to be random. Two types of dependent claims, main claims and by-claims, are defined, where every by-claim is induced by the main claim. The time of occurrence of a by-claim is later than that of its associate main claim and the time of delay for the occurrence of a by-claim is random. An integro-differential equations system for the Gerber–Shiu discounted penalty function is established using the auxiliary risk models. Both the system of Laplace transforms of the Gerber–Shiu discounted penalty functions and the Gerber–Shiu discounted penalty functions with zero initial surplus are obtained. From Lagrange interpolating theorem, we prove that the Gerber–Shiu discounted penalty function satisfies a defective renewal equation. Exact representation for the solution of this equation is derived through an associated compound geometric distribution. Finally, examples are given with claim sizes that have exponential and a mixture of exponential distributions.     

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

In 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.     

Omega model for a jump–diffusion process with a two-step premium rate

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

The Gerber-Shiu function for the compound Poisson Omega model with a three-step premium rate

Abstract

The compound Poisson Omega model is considered in the presence of a three-step premium rate. Firstly, the integral equations and the integro-differential equations for the Gerber-Shiu expected discounted penalty function are derived. Secondly, the integro-differential equations for the Gerber-Shiu expected discounted penalty function are determined in three different initial conditions. The results are then used to find the bankruptcy probability. Finally, the special cases where the claim size distribution is exponential be discussed in some detail in order to illustrate the effect of the model with three-step premium rate.     

On a discrete interaction risk model with delayed claims and stochastic incomes under random discount rates

In this paper, we study a discrete interaction risk model with delayed claims and stochastic incomes in the framework of the compound binomial model. A generalized Gerber-Shiu discounted penalty function is proposed to analyse this risk model in which the interest rates follow a Markov chain with finite state space. We derive an explicit expression for the generating function of this Gerber-Shiu discounted penalty function. Furthermore, we derive a recursive formula and a defective renewal equation for the original Gerber-Shiu discounted penalty function. As an application, the joint distributions of the surplus one period prior to ruin and the deficit at ruin, as well as the probabilities of ruin are obtained. Finally, some numerical illustrations from a specific example are also given.     

Optimal control strategies for dividend payments and capital injections in compound Markov binomial risk model with penalties for deficits

We consider the compound Markov binomial risk model. The company controls the amount of dividends paid to the shareholders as well as the capital injections in order to maximize the cumulative expected discounted dividends minus the discounted capital injections and the discounted penalties for deficits prior to ruin. We show that the optimal value function is the unique solution of an HJB equation, and the optimal control strategy is a two-barriers strategy given the current state of the Markov chain. We obtain some properties of the optimal strategy and the optimal condition for ruining the company. We offer a high-efficiency algorithm for obtaining the optimal strategy and the optimal value function. In addition, we also discuss the optimal control problem under a restriction of bounded dividend rates. Numerical results are provided to illustrate the algorithm and the impact of the penalties.     

On the Gerber–Shiu function with random discount rate

In this paper, we study the Gerber–Shiu (G-S) function for the classical risk model, in which the discount rate is generalized from a constant to a random variable. The discounted interest force accumulated process is modeled by a Poisson process and a Gaussian process for the G-S function. In terms of the standard techniques in ruin theory, we derive the integro-differential equation and the defective renewal equation satisfied by the G-S function. Then, the asymptotic formula for the G-S function is obtained using the renewal theory.     

Bayesian Dividend Optimization and Finite Time Ruin Probabilities

We consider the valuation problem of an (insurance) company under partial information. Therefore, we use the concept of maximizing discounted future dividend payments. The firm value process is described by a diffusion model with constant and observable volatility and constant but unknown drift parameter. For transforming the problem to a problem with complete information, we derive a suitable filter. The optimal value function is characterized as the unique viscosity solution of the associated Hamilton-Jacobi-Bellman equation. We state a numerical procedure for approximating both the optimal dividend strategy and the corresponding value function. Furthermore, threshold strategies are discussed in some detail. Finally, we calculate the probability of ruin in the uncontrolled and controlled situation.     

On the probability of ruin in a continuous risk model with two types of delayed claims

ABSTRACT

This article studies a risk model involving one type of main claims and two types of by-claims, which is an extension of the general risk model with delayed claims. We suppose that every main claim may not induce any by-claims or may induce one by-claim belonging to one of the two types of by-claims with a certain probability. In addition, assume that the by-claim and its associated main claim may occur at the same time and that the occurrence of the by-claim may be delayed. An integro-differential equation system for survival probabilities is derived by using two auxiliary risk models. The expression of the survival probability is obtained by applying Laplace transforms and Rouché theorem. Furthermore, we provide a method for solving the survival probability when the two by-claim amounts satisfy different exponential distributions. As a special case, an explicit expression of survival probability is given when all the claim amounts obey the same exponential distribution. Finally, numerical results are provided to examine the proposed method.     

The Discounted Moments of the Surplus After the Last Innovation Before Ruin Under the Dual Risk Model

□ This article's focus is on finding an explicit form of the discounted moments of the surplus at the time of the last jump before ruin for the compound Poisson dual risk model. For this purpose, we derive a non-homogeneous integro-differential equation, which is satisfied by the targeted quantity. To solve this equation, the general solution of the corresponding homogeneous equation and a particular solution of the non-homogeneous equation are obtained. Also, some additional results are provided, such as the defective distribution of the time to ruin and the Laplace transform of the time when the last jump before ruin happens.     

Asymptotic ruin probabilities of a two-dimensional renewal risk model with dependent inter-arrival times

Abstract

This article mainly considers the uniform asymptotics for the finite-time ruin probabilities of a two-dimensional renewal risk model with heavy-tailed claims. In this model, the two claim-number processes are arbitrarily dependent and each of them is generated by widely orthant dependent claim inter-arrival times. Two types of ruin are studied and for each type of ruin, an asymptotic formula for the finite-time ruin probability is established. These formulae possess a certain uniformity feature in the time horizon.     

The Ruin Probability of a Discrete-Time Risk Model with a One-Sided Linear Claim Process

In this article, the ruin probability is examined in a discrete time risk model with a constant interest rate, in which the dependent claims are assumed to have a one-sided linear structure. An explicit asymptotic formula is obtained for the ruin probability. Generalized Lundberg inequalities for the ruin probability are derived by martingale and inductive approaches.     

Uniformly asymptotic behavior for the tail probability of discounted aggregate claims in the time-dependent risk model with upper tail asymptotically independent claims

ABSTRACT

In this paper, we consider the tail behavior of discounted aggregate claims in a dependent risk model with constant interest force, in which the claim sizes are of upper tail asymptotic independence structure, and the claim size and its corresponding inter-claim time satisfy a certain dependence structure described by a conditional tail probability of the claim size given the inter-claim time before the claim occurs. For the case that the claim size distribution belongs to the intersection of long-tailed distribution class and dominant variation class, we obtain an asymptotic formula, which holds uniformly for all times in a finite interval. Moreover, we prove that if the claim size distribution belongs to the consistent variation class, the formula holds uniformly for all times in an infinite interval.     

Dependent Insurance Risk Model: Deterministic Threshold

This article considers a dependent insurance risk model. We assume that the inter-arrival time depends on the previous claim size through a deterministic threshold structure. Adjustment coefficient and Lundberg-type upper bound for the ruin probability are obtained. In case of exponential claim size, an explicit solution for the ruin probability is obtained by solving a system of ordinary delay differential equations. Some numerical results are included for illustration purposes.     

Tail behavior for the sum of two correlated classes of discounted aggregate claims in a time-dependent risk model

Consider a continuous-time risk model with two correlated classes of insurance business and a constant force of interest. Suppose that the correlation comes from a common shock, and that the claim sizes and inter-arrival times correspondingly form a sequence of random pairs, with each pair obeying a dependence structure. By assuming that the claim sizes are heavy tailed, a uniform tail asymptotic formula for the sum of the two correlated classes of discounted aggregate claims is obtained.     

Analysis of some ruin-related quantities in a Markov-modulated risk model

In this paper, we study the joint Laplace transform and probability generating function of some random quantities that occur in each environment state by the time of ruin in a Markov-modulated risk process. These quantities include the duration spent in each state, the number of claims and the aggregate amount of claims that occurred in each state by the time of ruin. Explicit formulae for the joint transforms, given the initial surplus, and the initial and terminal environment states, are expressed in terms of a matrix version of the scale function. Moments and covariances of these ruin-related quantities are obtained and numerical illustrations are presented. The joint transform of the duration spent in each state, the number of claims, and the aggregate amount of claims that occurred in each state by the time the surplus attains a certain level are also investigated.     

A surplus process involving a compound Poisson counting process and applications

Abstract

In this paper, we introduce a surplus process involving a compound Poisson counting process, which is a generalization of the classical ruin model where the claim-counting process is a homogeneous Poisson process. The incentive is to model batch arrival of claims using a counting process that is based on a compound distribution. This reduces the difficulty of modeling claim amounts and is consistent with industrial data. Recursive formula, some properties and relevant main ruin theory results are provided. Further, we consider applications involving zero-truncated negative binomial and zero-truncated binomial batch arrivals when the claim amounts follow exponential or Erlang distribution.