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

Numerical and Statistical Approximation of Stochastic Differential Equations with Non-Gaussian Measures

This monograph is based on methods and numerical tools from such fields as theory of stochastic differential equations (SDEs), stochastic modeling in computational physics, engineering and mathematical finance, statistical estimation methods, and Monte-Carlo type approximations.

EXAMPLES OF MEDICAL SOFTWARE AND HARDWARE EXPERT SYSTEMS FOR DYSFUNCTION ANALYSIS AND TREATMENT

Paper present the recent research in DMCS. The medical and biometric research projects are presented. One of the key element is an image acquisition and processing. The paper presents research of diagnostic application of voice analysis for stroke patients with speech dysfunction, as well as the method for diagnosing and monitoring the effectiveness of medical rehabilitation of patients with dysfunction of the cervical spine. Then the method for sudden cardiac death risk stratification is elaborated

O zastosowaniu drewnianych konstrukcji klejonych w budynkach biurowych

Due to the ongoing climate change, the issues currently in focus are the reduction of CO2 emissions into the atmosphere and sustainable construction. The search for ecological alternatives to traditional building structures that will reduce a building’s carbon footprint seems to be a desirable direction of modern construction development. At present, the first projects of office buildings that use cross-laminated timber as the main construction material are being completed, which can have a positive impact not only on the environment but also on the users of the building.Zmniejszenie emisji CO2 do atmosfery oraz budownictwo zrównoważone są obecnie zagadnieniami kluczowymi ze względu na postępujące zmiany klimatyczne. Poszukiwanie ekologicznych alternatyw dla tradycyjnych konstrukcji budynków, dzięki którym ślad węglowy będzie mniejszy, wydaje się pożądanym kierunkiem rozwoju współczesnego budownictwa. Powstają obecnie pierwsze realizacje budynków biurowych wykorzystujące jako główny materiał konstrukcyjny drewno klejone. Mogą one wpływać korzystnie nie tylko na środowisko, ale również na użytkowników obiektu

Computer simulation of a nonlinear model for electrical circuits with α-stable noise

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

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

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

Simulation and Chaotic Behavior of Alpha-stable Stochastic Processes

CONTENTS: Preliminary remarks; Brownian motion, poisson process, alpha-stable Levy motion; Computer simulation of alpha-stable random variables; Stochastic integration; Spectral representations of stationary processes; Computer approximations of continuous time processes; Examples of alpha-stable stochastic modelling; Convergence of approximate methods; Chaotic behaviour of stationary processes; Hierarchy of chaos for stable and ID stationary processes. Appendix - A guide to simulation.