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1
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Approximation of finite-dimensional distributions for integrals driven by α-stable Lévy motion

100%
Janicki A.
Applicationes Mathematicae
|
1998-1999
|
tom 25
|
nr 4
473-488
EN
We present a method of numerical approximation for stochastic integrals involving α-stable Lévy motion as an integrator. Constructions of approximate sums are based on the Poissonian series representation of such random measures. The main result gives an estimate of the rate of convergence of finite-dimensional distributions of finite sums approximating such stochastic integrals. Stochastic integrals driven by such measures are of interest in constructions of models for various problems arising in science and engineering, often providing a better description of real life phenomena than their Gaussian counterparts.
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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Bounds for the range of American contingent claim prices in the jump-diffusion model

64%
Janicki A., Wybraniec J.
Applicationes Mathematicae
|
2005
|
tom 32
|
nr 1
103-118
EN
The problem of valuation of American contingent claims for a jump-diffusion market model is considered. Financial assets are described by stochastic differential equations driven by Gaussian and Poisson random measures. In such setting the money market is incomplete, thus contingent claim prices are not uniquely defined. For different equivalent martingale measures different arbitrage free prices can be derived. The problem is to find exact bounds for the set of all possible prices obtained in this way. The paper extends and improves some results of [BJ00].
4
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Three-level discrete time Galerkin approximations for the non-stationary Navier-Stokes equation

63%
Janicki A.
Banach Center Publications
|
1978
|
tom 3
|
nr 1
167-174
5
Content available remote

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
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.
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