Approximation of stochastic differential equations driven by alpha-stable Levy motion

In this paper we present a result on convergence of approximate solutions of stochastic differential equations involving integrals with respect to alpha-stable Levy motion. We prove an appropriate weak limit theorem, which does not follow from known results on stability properties of stochastic differential equations driven by semimartingales. It assures convergence in law in the Skorokhod topology of sequences of approximate solutions and justifies discrete time schemes applied in computer simulations. An example is included in order to demonstrate that stochastic differential equations with jumps are of interest in constructions of models for various problems arising in science and engineering, often providing better description of real life phenomena than their Gaussian counterparts. In order to demonstrate the usefulness of our approach, we present computer simulations of a continuous time alpha-stable model of cumulative gain in the Duffie–Harrison option pricing framework.Stable distribution, Simulation, Stochastic differential equation (SDE), Option pricing

The Lamperti transformation for self-similar processes

In this paper we establish the uniqueness of the Lamperti transformation leading from self-similar to stationary processes, and conversely. We discuss alpha-stable processes, which allow to understand better the difference between the Gaussian and non-Gaussian cases. As a by-product we get a natural construction of two distinct alpha-stable Ornstein–Uhlenbeck processes via the Lamperti transformation for 0Lamperti transformation; Self-similar process; Stationary process; Stable distribution;

A new De Vylder type approximation of the ruin probability in infinite time

In this paper we introduce a generalization of the De Vylder approximation. Our idea is to approximate the ruin probability with the one for a different process with gamma claims, matching first four moments. We compare the two approximations studying mixture of exponentials and lognormal claims. In order to obtain exact values of the ruin probability for the lognormal case we use Pollaczeck-Khinchine formula. We show that the proposed 4-moment gamma De Vylder approximation works even better than the original one.Risk process; Ruin probability; De Vylder approximation; Pollaczeck-Khinchine formula;