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α-stable limits for multiple channel queues in heavy traffic

100%

Michna Z.

Applicationes Mathematicae

2003

tom 30

nr 1

55-68

We consider a sequence of renewal processes constructed from a sequence of random variables belonging to the domain of attraction of a stable law (1 < α < 2). We show that this sequence is not tight in the Skorokhod J₁ topology but the convergence of some functionals of it is derived. Using the structure of the sample paths of the renewal process we derive the convergence in the Skorokhod M₁ topology to an α-stable Lévy motion. This example leads to a weaker notion of weak convergence. As an application, we present limit theorems for multiple channel queues in heavy traffic. The convergence of the queue length process to a linear combination of α-stable Lévy motions is derived.

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

51%

Janicki A., Michna Z., Weron A.

Applicationes Mathematicae

1996-1997

tom 24

nr 2

149-168

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.

Ograniczanie wyników

2 Applicationes Mathematicae

2 Michna Z.

1 Janicki A.

1 Weron A.

1 2003

1 1997

Wyniki wyszukiwania

α-stable limits for multiple channel queues in heavy traffic

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