Florin Avram

ERLANGIAN APPROXIMATIONS FOR FINITE-HORIZON RUIN PROBABILITIES

For the Cram6r-Lundberg risk model with phase-type claims, it is shown that the probability of ruin before an independent phase-type time H coincides with the ruin probability in a certain Markovian fluid model and therefore has an matrix-exponential form. When H is exponential, this yields in particular a probabilistic interpretation of a recent result of Avram & Usabel. When H is Erlang, the matrix algebra takes a simple recursive form, and fixing the mean of H at T and letting the number of stages go to infinity yields a quick approximation procedure for the probability of ruin before time T. Numerical examples are given, including a combination with extrapolation.

Phase-type approximations to finite-time ruin probabilities in the Sparre-Andersen and stationary renewal risk models

The present paper extends the “Erlangization” idea introduced by Asmussen, Avram, and Usabel (2002) to the Sparre-Andersen and stationary renewal risk models. Erlangization yields an asymptotically-exact method for calculating finite time ruin probabilities with phase-type claim amounts. The method is based on finding the probability of ruin prior to a phase-type random horizon, independent of the risk process. When the horizon follows an Erlang-l distribution, the method provides a sequence of approximations that converges to the true finite-time ruin probability as l increases. Furthermore, the random horizon is easier to work with, so that very accurate probabilities of ruin are obtained with comparatively little computational effort. An additional section determines the phase-type form of the deficit at ruin in both models. Our work exploits the relationship to fluid queues to provide effective computational algorithms for the determination of these quantities, as demonstrated by the numerical examples.

On the valuation of constant barrier options under spectrally one-sided exponential Lévy 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 Levy model, and uses it to implement of Carrs approximation for the value of the American put under this model. Simple analytic approximations for the exercise boundary and option value are obtained.

Explicit Solutions for Variational Problems in the Quadrant

We study a variational problem (VP) that is related to semimartingale reflecting Brownian motions (SRBMs). Specifically, this VP appears in the large deviations analysis of the stationary distribution of SRBMs in the d-dimensional orthant Rd+. When d=2, we provide an explicit analytical solution to the VP. This solution gives an appealing characterization of the optimal path to a given point in the quadrant and also provides an explicit expression for the optimal value of the VP. For each boundary of the quadrant, we construct a “cone of boundary influence”, which determines the nature of optimal paths in different regions of the quadrant. In addition to providing a complete solution in the 2-dimensional case, our analysis provides several results which may be used in analyzing the VP in higher dimensions and more general state spaces.

Finite time ruin probabilities with one Laplace inversion

Abstract 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 well.

Robustness of the R / S Statistic for Fractional Stable Noises

The R / S statistic is used to detect long-range dependence in a time series and to estimate its intensity. One of its virtues is robustness against different distributions. We show here that the R / S statistic continues to be robust if the time series is a moving average with long-range dependence with innovations that are in the domain of attraction of an infinite variance stable process.

Hölder’s Inequality for Functions of Linearly Dependent Arguments

Functions with values in

AN EXAMINATION OF THE SIGN AND VOLATILITY SWITCHING ARCH MODELS UNDER ALTERNATIVE DISTRIBUTIONAL ASSUMPTIONS

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Russian and American put options under exponential phase-type Lévy models

on a torus, in

On the optimal dividend problem for a spectrally negative Levy process

L_p ( - \infty , + \infty )