Classical Risk Process Research Articles

Optimal control of risk exposure, reinsurance and investments for insurance portfolios

We consider an insurance company whose insurance business follows a diffusion perturbated classical risk process. Assets can be invested in a risk-free asset or in a risky one, the return of the latter follows an independent diffusion perturbated classical risk process (in finance it is called a jump-diffusion process). The company can dynamically control the proportion of the insurance business that is ceded to reinsurers, and in addition it can dynamically cover parts of its remaining business using excess of loss reinsurance. It can also dynamically control the proportion of the assets it invests in the risky asset. We seek to find the controls that maximize expected utility of assets at a terminal time. We do so for three different utility functions. Let T be a fixed time and Ty be the time of ruin, i.e. the first time assets become nonpositive. Then the utility functions and terminal times are: 1.U(y)=(max{0,y})c, 0<c<1and terminal time is T∧Ty.2.U(y)=ln ywhen y>0 and U(y)=−∞ when y≤0. Terminal time is T.3.U(y)=−e−cy, c>0, and terminal time is T.It turns out that in spite of the complexity of the problem, provided there are no complicating constraints on the controls, the problems have rather simple solutions. Our vehicle is the Hamilton–Jacobi–Bellmann equations, and we seek to verify that the suggested solutions are in fact optimal among a fairly large easily verifiable admissible class of controls. We also discuss quantitative and qualitative aspects of the solutions.

Method of Successive Approximations for Solving Integral Equations of the Theory of Risk Processes

A model of a classical risk process describing the evolution of an insurance company's capital is generalized. Integral equations for the bankruptcy probability are derived. The method of successive approximations is used to solve these equations.

Asymptotics of Ruin Probabilities for Risk Processes under Optimal Reinsurance and Investment Policies: The Large Claim Case

In a classical risk process reinsurance and investment can be chosen at any time. We find the Lundberg exponent and the Cramer–Lundberg approximation for the ruin probability under the optimal strategy in the case where no exponential moments for the claim size distribution exist. We also show that the optimal strategies converge.

Ruin probabilities for a risk process with stochastic return on investments

In this paper, we consider a risk process with stochastic return on investments. The basic risk process is the classical risk process while the return on the investment generating process is a compound Poisson process plus a Brownian motion with positive drift. We obtain an integral equation for the ultimate ruin probability which is twice continuously differentiable under certain conditions. We then derive explicit expressions for the lower bound for the ruin probability. We also study a joint distribution related to exponential functionals of Brownian motion which is required in the derivations of the explicit expressions for the lower bound.

Some Results of Ruin Probablility for the Classical Risk Process

Some Results of Ruin Probablility for the Classical Risk Process

The time of ruin, the surplus prior to ruin and the deficit at ruin for the classical risk process perturbed by diffusion

The paper studies the joint distribution of the time of ruin, the surplus prior to ruin and the deficit at ruin for the classical risk process that is perturbed by diffusion. We prove that the expected discounted penalty satisfies an integro-differential equation of renewal type, the solution of which can be expressed as a convolution formula. The asymptotic behaviour of the expected discounted penalty as the initial capital tends to infinity is discussed.

The joint density function of three characteristics on jump-diffusion risk process

In this paper, we consider the jump-diffusion risk process, i.e., the classical risk process that is perturbed by diffusion. We derive the explicit expression for the joint density function of three characteristics: the time of ruin, the surplus immediately before ruin, and the deficit at ruin. By using the explicit joint density function, we extend some Dickson’s [Insurance: Mathematics and Economics 11 (1992) 191] results to the jump-diffusion risk process. We also obtain the distribution of the time that the negative surplus first reaches the level zero.

EXPLICIT EXPRESSIONS FOR SOME DISTRIBUTIONS RELATED TO RUIN PROBLEMS

The classical risk process that is perturbed by diffusion is studied. The explicit expressions for the ruin probability and the surplus distribution of the risk process at the time of ruin are obtained when the claim amount distribution is a finite mixture of exponential distributions or a Gamma (2, α) distribution.

Some results of ruin probability for the classical risk process

The computation of ruin probability is an important problem in the collective risk theory. It has applications in the fields of insurance, actuarial science, and economics. Many mathematical models have been introduced to simulate business activities and ruin probability is studied based on these models. Two of these models are the classical risk model and the Cox model. In the classical model, the counting process is a Poisson process and in the Cox model, the counting process is a Cox process. Thorin (1973) studied the ruin probability based on the classical model with the assumption that random sequence followed the Γ distribution with density function f(x)=x1β−1β1βΓ(1/β)e−xβ, x&gt;0, where β&gt;1. This paper studies the ruin probability of the classical model where the random sequence follows the Γ distribution with density function f(x)=αnΓ(n)xn−1e−αx, x&gt;0, where α&gt;0 and n≥2 is a positive integer. An intermediate general result is given and a complete solution is provided for n=2. Simulation studies for the case of n=2 is also provided.

Ruin problem for a class of risk processes perturbed by diffusion

In this paper, a class of risk processes perturbed by diffusion are considered. The Lundberg inequalities for the ruin probability are obtained. The size of the Lundberg exponents for different kinds of risk model is compared. The numerical illustration for the impact of the parameters on the ruin probability is given.

Some Results for Classical Risk Process with Stochastic Return on Investments

In this paper, we discuss the classical risk process with stochastic return on investment. We prove some properties of the ruin probability, the supremum distribution before ruin and the surplus distribution at the time of ruin and derive the integro-differential equations satisfied by these distributions respectively.

On Cramér-like asymptotics for risk processes with stochastic return on investments

We consider a classical risk process compounded by another independent process. Both of these component processes are assumed to be Levy processes. We show asymptotically that as initial capital $y$ increases the ruin probability will essentially behave as $y^{-\kappa}$, where $\kappa$ depends on one of the component processes.

Spectrally negative Lévy processes with applications in risk theory

In this paper, results on spectrally negative Levy processes are used to study the ruin probability under some risk processes. These processes include the compound Poisson process and the gamma process, both perturbed by diffusion. In addition, the first time the risk process hits a given level is also studied. In the case of classical risk process, the joint distribution of the ruin time and the first recovery time is obtained. Some results in this paper have appeared before (e.g., Dufresne and Gerber (1991), Gerber (1990), dos Reis (1993)). We revisit them from the Levy process theory's point of view and in a unified and simple way.

Spectrally negative Lévy processes with applications in risk theory

In this paper, results on spectrally negative Lévy processes are used to study the ruin probability under some risk processes. These processes include the compound Poisson process and the gamma process, both perturbed by diffusion. In addition, the first time the risk process hits a given level is also studied. In the case of classical risk process, the joint distribution of the ruin time and the first recovery time is obtained. Some results in this paper have appeared before (e.g., Dufresne and Gerber (1991), Gerber (1990), dos Reis (1993)). We revisit them from the Lévy process theory's point of view and in a unified and simple way.

A decomposition of the ruin probability for the risk process perturbed by diffusion

In this paper, we consider the ruin probabilities (caused by oscillation or by a claim) of the classical risk process perturbed by diffusion and the risk process with return on investments. We will prove their twice continuous differentiability and derive the integro-differential equations satisfied by them. We will present the explicit expressions for them when the claims are exponentially distributed.

Some distributions for classical risk process that is perturbed by diffusion

In this paper we discuss the classical risk process that is perturbed by diffusion. We prove some properties of the supremum distribution of the risk process before ruin when ruin occurs and the surplus distribution at the time of ruin. We present the simple and explicit expression for these distributions when the claims are exponentially distributed.

A generalization of risk model pertured by diffusion

In this paper, the classical risk process perturbed by diffusion is generalized by allowing for “size fluctuation” and the ruin probability for this new model is discussed.

A note on the Taylor series expansions for multivariate characteristics of classical risk processes

The series expansion introduced by Frey and Schmidt (1996) [Taylor Series expansion for multivariate characteristics of classical risk processes. Insurance: Mathematics and Economics 18, 1–12.] constitutes an original approach in approximating multivariate characteristics of classical ruin processes, specially ruin probabilities within finite time with certain surplus prior to ruin and severity of ruin. This approach can be considered alternative to inversion of Laplace transforms for particular claim size distributions [Gerber, H., Goovaerts, M., Kaas, R., 1987. On the probability and severity of ruin. ASTIN Bulletin 17(2), 151–163; Dufresne, F., Gerber, H., 1988a. The probability and severity of ruin for combinations of exponential claim amount distributions and their translations. Insurance: Mathematics and Economics 7, 75–80; Dufresne, F., Gerber, H., 1988b. The surpluses immediately before and at ruin, and the amount of the claim causing ruin. Insurance: Mathematics and Economics 7, 193–199.] or discretization of the claim size and time [Dickson, C., 1989. Recursive calculation of the probability and severity of ruin. Insurance: Mathematics and Economics 8, 145–148; Dickson, C., Waters, H., 1992. The probability and severity of ruin in finite and infinite time. ASTIN Bulletin 22(2), 177–190; Dickson, C., 1993. On the distribution of the claim causing ruin. Insurance: Mathematics and Economics 12, 143–154.] applying the so-called Panjer’s recursive algorithm [Panjer, H.H., 1981. Recursive calculation of a family of compound distributions. ASTIN Bulletin 12, 22–26.]. We will prove that the recursive relation involved in the calculations of the the nth derivative with respect to λ – average number of claims in the time unit – of the multivariate finite time ruin probability (developed in the original paper by Frey and Schmidt (1996) can be simplified. The cited simplification leads to a substantial reduction in the number of multiple integrals used in the calculations and makes the series expansion approach more appealing for practical implementation.

Level-Crossing Properties of the Risk Process

For the classical risk process R(t) that is linear increasing with slope 1 between downward jumps of i.i.d. random sizes at the points of a homogeneous Poisson process we consider the level-crossing process C(x) = (L(x), (Ai (x), Bi (x))1≤ i ≤ L ( x )), where L(x) is the number of jumps from (x, ∞) to (−∞, x] and Ai (x) (Bi (x)) are the distances from x to R(t) after (before) the ith jump of this kind. It is shown that if R(·) has a drift toward infinity, C(·) is a stationary Markov process; its transition probabilities are determined. As an application we derive the expected value E(L(x)L(x + y)).

On a Class of Renewal Risk Processes

In this paper I show how methods that have been applied to derive results for the classical risk process can be adapted to derive results for a class of risk processes in which claims occur as a renewal process. In particular, claims occur as an Erlang process. I consider the problem of finding the survival probability for such risk processes and then derive expressions for the probability and severity of ruin and for the probability of absorption by an upper barrier. Finally, I apply these results to consider the problem of finding the distribution of the maximum deficit during the period from ruin to recovery to surplus level 0.