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

100%
Janicki A., Michna Z., Weron A.
Applicationes Mathematicae
|
1996-1997
|
tom 24
|
nr 2
149-168
EN
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.
2
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Computer simulation of a nonlinear model for electrical circuits with α-stable noise

100%
Janicki A.
Applicationes Mathematicae
|
1995-1996
|
tom 23
|
nr 1
95-105
EN
The aim of this paper is to apply the appropriate numerical, statistical and computer techniques to the construction of approximate solutions to nonlinear 2nd order stochastic differential equations modeling some engineering systems subject to large random external disturbances. This provides us with quantitative results on their asymptotic behavior.
3
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Computer-aided modeling and simulation of electrical circuits with α-stable noise

88%
Weron A.
Applicationes Mathematicae
|
1995-1996
|
tom 23
|
nr 1
83-93
EN
The aim of this paper is to demonstrate how the appropriate numerical, statistical and computer techniques can be successfully applied to the construction of approximate solutions of stochastic differential equations modeling some engineering systems subject to large disturbances. In particular, the evolution in time of densities of stochastic processes solving such problems is discussed.
4
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On the book ``An Introduction to Differential Equations: Stochastic Modeling, Methods and Analysis'' by A.G.Ladde and G.S.Ladde

63%
Jurlewicz A.
Mathematica Applicanda
|
2015
|
tom 43
|
nr 2
PL
Niniejsza książka stanowi kontynuację podręcznika, tych samych autorów, przedstawiającego tematykę równań różniczkowych. Tom 1. (Deterministic Modeling, Methods and Analysis) dotyczył teorii klasycznych, natomiast omawiany tu tom 2. prezentuje ideę równań różniczkowych stochastycznych i ich zastosowania w modelowaniu matematycznym. Książka adresowana jest głównie do studentów i doktorantów kierunków interdyscyplinarnych.
EN
The book under review presents advanced tools of stochastic calculus and stochastic differential equations of Ito type, illustrated by several problems and applications. It is a continuation of Volume 1: Deterministic Modeling, Methods and Analysis. It is addressed to interdisciplinary graduate/undergraduate students and to interdisciplinary young researchers.
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