Approximation of stochastic differential equations driven by α-stable Lévy motion

Volume 24 / 1997

Aleksander Janicki, Zbigniew Michna, Aleksander Weron Applicationes Mathematicae 24 (1997), 149-168 DOI: 10.4064/am-24-2-149-168

Abstract

In this paper we present a result on convergence of approximate solutions of stochastic differential equations involving integrals with respect to α-stable Lévy 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 α-stable model of cumulative gain in the Duffie-Harrison option pricing framework.

Authors

  • Aleksander Janicki
  • Zbigniew Michna
  • Aleksander Weron

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