Perturbed Risk Process Research Articles

Sub-optimal investment for insurers

We consider the investment problem for a non-life insurance company seeking to minimize the ruin probability. Its reserve is described by a perturbed risk process possibly correlated with the financial market. Assuming exponential claim size, the Hamilton-Jacobi-Bellman equation reduces to a first order nonlinear ordinary differential equation, which seems hard to solve explicitly. We study the qualitative behavior of its solution and determine the Cramér-Lundberg approximation. Moreover, our approach enables to find very naturally that the optimal investment strategy is not constant. Then, we analyze how much the company looses by adopting sub-optimal constant (amount) investment strategies.

Ruin Probabilities in Perturbed Risk Process with Stochastic Investment and Force of interest

This work considers a perturbed risk process with investment, where the investments are either into invested in risky and risk-less assets. A third order differential equation for the ruin probability is derived from the resulting integrodifferential equation. This equation is further decomposed into two equations describing the contributions of the claim and oscillation to the ruin probability. These two equations are solved separately using suitable transformations as well as theory of Kummer confluence hypergeometric equations.We further investigated these solutions and were able to conclude that the higher the fraction of investment into risky assets, the lower the ruin probability, provided all other parameters are kept constant. Keywords: Risk Reserve; Ruin Probability; Interest; Stochastic Investment; Exponential distribution; Kummer hyper-geometric equation AMS 2010 Mathematics Subject Classification : 60J25; 60J60

Ruin time and aggregate claim amount up to ruin time for the perturbed risk process

We consider the classical Sparre-Andersen risk process perturbed by a Wiener process, and study the joint distribution of the ruin time and the aggregate claim amounts until ruin by determining its Laplace transform. This is first done when the claim amounts follow respectively an exponential/Phase-type distribution, in which case we also compute the distribution of recovery time and study the case of a barrier dividend. Then the general distribution is considered when ruin occurs by oscillation, in which case a renewal equation is derived.

The maximum severity of ruin in a perturbed risk process with Markovian arrivals

In this paper, we present a result on the distribution of the maximum severity of ruin in a perturbed risk process with Markovian arrivals. We show that the distribution of the maximum severity of ruin is closely related to the distributions of one-sided and two-sided passage times.

Four approaches to compute the probability of ruin in the compound Poisson risk process with diffusion

This article compares four different approaches for computing the probability of ruin of the insurer compound Poisson surplus process with an additive Wiener perturbation. The first method is based on the saddlepoint approximation of asymptotic analysis. The second method is based on recursive upper and lower approximations and it is a new approach for the perturbed risk process. The third method is the fast Fourier transform. The last method is Monte Carlo importance sampling. A numerical study illustrates the high accuracy of these four methods.

Erlangian approximation to finite time ruin probabilities in perturbed risk models

In this paper, we consider a class of perturbed risk processes that have an underlying Markov structure, including Markov-modulated risk processes, and Sparre–Andersen risk processes when both inter-claim times and claim sizes are phase-type. We apply the Erlangization method to the risk process in the class in order to obtain an accurate approximation of the finite time ruin probability. In addition, we develop an efficient recursive procedure by recognizing a repeating structure in the probability matrices we work with. We believe the present work is among the first to either compute or approximate finite time ruin probabilities in the perturbed risk model.

Perturbed Risk Processes Analyzed as Fluid Flows

In this article, we present a unified phase-type approach for calculating ruin probabilities for a class of perturbed risk processes, which includes the perturbed Sparre–Andersen process with phase-type interclaim time and claim size distributions, the perturbed Markov-modulated risk process, and the MAP/PH risk process as well. The key to the solution method is the identification for a given risk process of a parallel Markov-modulated fluid flow. The resulting group of perturbed risk processes amenable to this approach we refer to as the Markov-modulated fluid-flow equivalent (MFE) class, as the ability to find an equivalent fluid flow is the only limiting factor in our analysis. The primary contributions of this work are 1) the unified, tractable, phase-type structures for the maximal aggregate loss and ladder height distributions for all variants of perturbed risk processes belonging to the MFE class, and 2) the explicit formulas for the ruin probability for all variants in this class. The full breadth of the models is illustrated through a diverse range of examples.

A perturbed risk process compounded by a geometric Brownian motion with a dividend barrier strategy

In this paper, we consider a perturbed risk process (in which the inter-occurrence times are generalized Erlang( n)-distributed) compounded by a geometric Brownian motion. Integro-differential equations with certain boundary conditions for the moment-generating function and the mth moment of the present value of all dividends until ruin are derived. We also derive integro-differential equations with boundary conditions for the Gerber–Shiu function. Some special cases are considered in details.

On the surplus prior to ruin in the perturbed classical risk process

PurposeThe purpose of this article is to consider the classical risk model that is perturbed by a Brownian motion process. The article derives explicit formulas for the joint and marginal probability density functions of the surplus prior to ruin and the deficit at ruin.Design/methodology/approachThis article first extends the dual argument to probabilistically explain the symmetry between the two random variables related to the so‐called modified ladder height. Then the paper uses renewal arguments to derive the joint distribution of the surplus prior to ruin and the deficit at ruin.FindingsThe study derived an explicit formula for the undiscounted joint density in the perturbed risk model that is directly parallel to formula (3.2) for the classical risk model. The formula clearly shows that in a perturbed risk process, when ruin is caused by a claim, the p.d.f. of the surplus prior to ruin is continuous. In addition, shows that when the claim sizes follow a phase‐type distribution, all the relevant quantities can be conveniently computed.Originality/valueThe dual argument used in this article is novel. The formula first clearly shows that in the perturbed risk model, the p.d.f. of the surplus prior to ruin is continuous. When claim sizes are phase‐type, the formulas can be conveniently computed.

On the expected discounted penalty function for a perturbed risk process driven by a subordinator

The Expected Discounted Penalty Function (EDPF) was introduced in a series of now classical papers [Gerber, H.U., Shiu, E.S.W., 1997. The joint distribution of the time of ruin, the surplus immediately before ruin, and the deficit at ruin. Insurance: Math. Econ. 21, 129–137; Gerber, H.U., Shiu, E.S.W., 1998a. On the time value of ruin. N. Am. Actuar. J. 2 (1), 48–78; Gerber, H.U., Shiu, E.S.W., 1998b. Pricing perpetual options for jump processes. N. Am. Actuar. J. 2 (3), 101–112]. Later, Gerber and Landry [Gerber, H.U., Landry, B., 1998. On a discounted penalty at ruin in a jump–diffusion and the perpetual put option. Insurance: Math. Econ. 22, 263–276] extended the concept to the perturbed case. Recent papers have extended these results in more general settings [for instance Tsai and Willmot [Tsai, C.C.L., Willmot, G.E., 2002. A generalized defective renewal equation for the surplus process perturbed by diffusion. Insurance: Math. Econ. 30, 51–66], Li and Lu [ Li, S., Lu, Y., 2005. On the expected discounted penalty function for two classes of risk processes. Insurance: Math. Econ. 36, 179–193] and Li and Garrido [ Li, S., Garrido, J., 2005. On the Gerber–Shiu functions in a Sparre–Andersen risk model perturbed by diffusion. Scand. Actuar. J. 161–186]]. In this note we present yet another generalization that has not been considered before in the literature. We present a perturbed risk process with a subordinator as the model for the aggregate claims. We generalize existing results [Tsai, C.C.L., Willmot, G.E., 2002. A generalized defective renewal equation for the surplus process perturbed by diffusion. Insurance: Math. Econ. 30, 51–66] on the EDPF for the subordinator case.

Ruin probabilities for competing claim processes

Let C1, C2,…,Cm be independent subordinators with finite expectations and denote their sum by C. Consider the classical risk process X(t) = x + ct - C(t). The ruin probability is given by the well-known Pollaczek–Khinchin formula. If ruin occurs, however, it will be caused by a jump of one of the subordinators Ci. Formulae for the probability that ruin is caused by Ci are derived. These formulae can be extended to perturbed risk processes of the type X(t) = x + ct - C(t) + Z(t), where Z is a Lévy process with mean 0 and no positive jumps.

Ruin probabilities for competing claim processes

Let C 1, C 2,…,C m be independent subordinators with finite expectations and denote their sum by C. Consider the classical risk process X(t) = x + ct - C(t). The ruin probability is given by the well-known Pollaczek–Khinchin formula. If ruin occurs, however, it will be caused by a jump of one of the subordinators C i . Formulae for the probability that ruin is caused by C i are derived. These formulae can be extended to perturbed risk processes of the type X(t) = x + ct - C(t) + Z(t), where Z is a Lévy process with mean 0 and no positive jumps.

Ruin probabilities and decompositions for general perturbed risk processes

We study a general perturbed risk process with cumulative claims modelled by a subordinator with finite expectation, with the perturbation being a spectrally negative Levy process with zero expectation. We derive a Pollaczek-Hinchin type formula for the survival probability of that risk process, and give an interpretation of the formula based on the decomposition of the dual risk process at modified ladder epochs.

Asymptotic results for perturbed risk processes with delayed claims

The object of this paper is the study of some asymptotic properties of the perturbed risk process with delayed claims ( X t ), which is the sum of a Brownian motion with drift and a shot-noise whose underlying point process is a doubly stochastic Poisson process. More in particular, under suitable hypotheses, we show that ( X t ) satisfies a large deviation principle, and we give asymptotic estimates of the corresponding ruin probabilities. Moreover, we introduce two suitable processes ( L t ( X) ) and ( R t ( X) ), which can be seen as simplified versions of ( X t ), and we show some inequalities between the rate function and the Lundberg parameter concerning ( X t ), and the rate functions and the Lundberg parameters concerning ( L t ( X) ) and ( R t ( X) ), respectively.

A Diffusion Perturbed Risk Process with Stochastic Return on Investments

A general risk model that allows for stochastic return on investments as well as perturbation by diffusion is studied. Integro-differential equations for the distributions of the time of ruin, the surplus prior to ruin and the deficit at ruin of this model are established. In particular, we consider a diffusion perturbed risk model with interest force in details.

Cramér–Lundberg approximation for nonlinearly perturbed risk processes

An extension of the classical Cramér–Lundberg approximation for ruin probabilities to a model of nonlinearly perturbed risk processes is presented. We introduce correction terms for the Cramér–Lundberg and diffusion type approximations, which provide the right asymptotic behaviour of relative errors in a perturbed model. The dependence of these correction terms on relations between the rate of perturbation and the speed of growth of an initial capital is investigated. Various types of perturbations of risk processes are discussed. The results are based on a new type of exponential asymptotics for perturbed renewal equations.