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 finit 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 finit and infinit 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 – average number of claims in the time unit – of the multivariate finit time ruin probability (developed in the original paper by Frey and Schmidt (1996) can be simplified The cited simplificatio 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 implementationFinite time ruin probability; Surplus prior to ruin; Severity of ruin; Series expansion; Recursive methods;

Calculating multivariate ruin probabilities via Gaver–Stehfest inversion technique.

Multivariate characteristics of risk processes are of high interest to academic actuaries. In such models, the probability of ruin is obtained not only by considering initial reserves u but also the severity of ruin y and the surplus before ruin x. This ruin probability can be expressed using an integral equation that can be efficiently solved using the Gaver–Stehfest method of inverting Laplace transforms. This approach can be considered to be an alternative to recursive methods previously used in actuarial literatureMultivariate ultimate ruin probability; Laplace transform; Integral equations; Numerical methods;

The Gerber-Shiu expected discounted penalty-reward function under an affine jump-diffusion model.

We provide a unified analytical treatment of first passage problems under an affine state-dependent jump-diffusion model (with drift and volatility depending linearly on the state). Our proposed model, that generalizes several previously studied cases, may be used for example for obtaining probabilities of ruin in the presence of interest rates under the rational investement strategies proposed by Berk & Green (2004)First passage problems; Risk process; Stochastic rates of interest; Ruin with interest; Affine jump-diffusion models; Penalty/reward functions at ruin;

A numerical method for the expected penalty–reward function in a Markov-modulated jump–diffusion process.

A generalization of the Cramér–Lundberg risk model perturbed by a diffusion is proposed. Aggregate claims of an insurer follow a compound Poisson process and premiums are collected at a constant rate with additional random fluctuation. The insurer is allowed to invest the surplus into a risky asset with volatility dependent on the level of the investment, which permits the incorporation of rational investment strategies as proposed by Berk and Green (2004). The return on investment is modulated by a Markov process which generalizes previously studied settings for the evolution of the interest rate in time. The Gerber–Shiu expected penalty–reward function is studied in this context, including ruin probabilities (a first-passage problem) as a special case. The second order integro-differential system of equations that characterizes the function of interest is obtained. As a closed-form solution does not exist, a numerical procedure based on the Chebyshev polynomial approximation through a collocation method is proposed. Finally, some examples illustrating the procedure are presentedExpected penalty–reward function; Markov-modulated process; Jump–diffusion process; Volterra integro-differential system of equations;

Finite time ruin probabilities with one Laplace inversion.

In this work we present an explicit formula for the Laplace transform in time of the finite time ruin probabilities of a classical Levy model with phase-type claims. Our result generalizes the ultimate ruin probability formula of Asmussen and Rolski [IME 10 (1991) 259]—see also the analog queuing formula for the stationary waiting time of the M/Ph/1 queue in Neuts [Matrix-geometric Solutions in Stochastic Models: An Algorithmic Approach. Johns Hopkins University Press, Baltimore, MD, 1981]—and it considers the deficit at ruin as wellFinite-time ruin probability; Phase-type distribution; Deficit at ruin; Lundberg’s equation; Laplace transform;

On the valuation ofconstant barrier options under spectrally one-sided exponential L&evy models and Carr’s approximation for American puts.

This paper provides a general framework for pricing options with a constant barrier under spectrally one-sided exponential L&evy model, and uses it to implement ofCarr’s approximation for the value of the American put under this model. Simple analytic approximations for the exercise boundary and option value are obtained. c 2002 Elsevier Science B.V. All rights reservedAmerican options; Perpetual approximation; Spectrally negative exponential L&evy process;

Applications to risk theory of a Monte Carlo multiple integration method.

Evaluation of multiple integrals is a commonly encountered problem in risk theory, specially in ruin probability. Using Monte Carlo simulation we obtain an unbiased and consistent point estimator, and also confidence intervals as approximations of a special case of multiple integral frequently used in risk theory. The variance reduction achieved compared to straight simulation and some specific properties make this approach interesting when approximating ruin probabilitiesMonte Carlo multiple integration; Variance reduction; Convolutions; Ruin probability;