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1
Content available Stochastic models of the slow/fast type of atrioventricular nodal reentrant tachycardia and tachycardia with conduction aberration
Jackowska-Zduniak Beata
International Journal of Applied Mathematics and Computer Science
|
2022
|
Vol. 32, no. 3
429--440
EN
Models are proposed to describe the heart’s action potential. A system of stochastic differential equations is used to recreate pathological behaviour in the heart such as atrioventricular nodal reentrant tachycardia (AVNRT) and also AVNRT with conduction aberration. Part of the population has abnormal accessory pathways: fast and slow. An additional pathway is not always induced, since the deterministic model is not proper due to a stochasticity in this process. Introduction of a stochastic term allows modelling a pre-excitation perturbation (such as unexpected excitation by premature contractions in atrium (PAC)) which triggers the mechanism of AVNRT. Also, a system of AVNRT with additional conduction aberration, which is a rare type of arrhythmia, is considered. The aim of this work is to propose a mathematical model superior to the deterministic one that recreates this disease better and allows understanding its mechanism and physical dependencies, which may help to propose a new therapy of AVNRT. Results are illustrated with numerical solutions.
2
Content available remote A Note on Existence of Global Solutions and Invariant Measures for Jump Sdes with Locally One-Sided Lipschitz Drift
Majka Mateusz B.
Probability and Mathematical Statistics
|
2020
|
Vol. 40, Fasc. 1
37--55
EN
We extend some methods developed by Albeverio, Brzeźniak and Wu and we show how to apply them in order to prove existence of global strong solutions of stochastic differential equations with jumps, under a local one-sided Lipschitz condition on the drift (also known as a monotonicity condition) and a local Lipschitz condition on the diffusion and jump coefficients, while an additional global one-sided linear growth assumption is satisfied. Then we use these methods to prove existence of invariant measures for a broad class of such equations.
3
Content available Review of stochastic differential equations in statistical arbitrage pairs trading
Endres Sylvia
Managerial Economics
|
2019
|
Vol. 20, no. 2
71--118
EN
The use of stochastic differential equations offers great advantages for statistical arbitrage pairs trading. In particular, it allows the selection of pairs with desirable properties, e.g., strong mean-reversion, and it renders traditional rules of thumb for trading unnecessary. This study provides an exhaustive survey dedicated to this field by systematically classifying the large body of literature and revealing potential gaps in research. From a total of more than 80 relevant references, five main strands of stochastic spread models are identified, covering the ‘Ornstein–Uhlenbeck model’, ‘extended Ornstein–Uhlenbeck models’, ‘advanced mean-reverting diffusion models’, ‘diffusion models with a non-stationary component’, and ‘other models’. Along these five main categories of stochastic models, we shed light on the underlying mathematics, hereby revealing advantages and limitations for pairs trading. Based on this, the works of each category are further surveyed along the employed statistical arbitrage frameworks, i.e., analytic and dynamic programming approaches. Finally, the main findings are summarized and promising directions for future research are indicated.
4
Content available remote Weak convergence of a numerical scheme for stochastic differential equations
Aguilera E., Fierro R.
Probability and Mathematical Statistics
|
2017
|
Vol. 37, Fasc. 1
201--215
EN
In this paper a numerical scheme approximating the solution to a stochastic differential equation is presented. On bounded subsets of time, this scheme has a finite state space, which allows us to decrease the round-off error when the algorithm is implemented. At the same time, the scheme introduced turns out locally consistent for any step size of time. Weak convergence of the scheme to the solution of the stochastic differentia equation is shown.
5
Content available remote Cross-variation of Young integral with respect to long-memory fractional Brownian motions
Nourdin I., Zintout R.
Probability and Mathematical Statistics
|
2016
|
Vol. 36, Fasc. 1
35--46
EN
We study the asymptotic behaviour of the cross-variation of two-dimensional processes having the form of a Young integral with respect to a fractional Brownian motion of index H > 1/2 . When H is smaller than or equal to 3/4 , we show asymptotic mixed normality. When H is stricly greater than 3/4 , we obtain a limit that is expressed in terms of the difference of two independent Rosenblatt processes.
6
Produkcja energii elektrycznej i ciepła z odnawialnych nośników w warunkach ryzyka
Sowiński J., Pydych T., Tomaszewski R., Wachtarczyk A.
Rynek Energii
|
2015
|
Nr 2
59--65
PL
W ostatnich latach znacznie zwiększa się wykorzystanie odnawialnych nośników energii. Tendencja ta związana jest głównie z wyczerpywaniem się zasobów paliw kopalnych oraz z problemami oddziaływania konwencjonalnej energetyki na środowisko, głównie w zakresie emisji gazów cieplarnianych. Zaprezentowano statystyki, cyklicznie publikowanej przez GUS, a dotyczące produkcji energii elektrycznej i ciepła z odnawialnych nośników energii w latach 2003-2013. Zamodelowano przebiegi produkcji energii elektrycznej i ciepła stochastycznymi równaniami różniczkowymi. Rozwiązanie równań metodą Eulera umożliwia symulację przebiegów w horyzoncie średnioterminowym. Modelowanie przebiegów w postaci równań stochastycznych pozwala na wprowadzenie do analizy ryzyka.
EN
The use of renewable energy sources has been significantly increased since a few last years. The tendency is generally connected with limited resources of fossil fuels and with environmental impacts of power engineering, especially emissions of greenhouse gases. Statistical data of generation of electricity and heat from RES, frequently published by GUS, are presented. On that basis some medium-term simulations of electricity generation and heat production from RES are presented using stochastic differential equations (SDE) and Euler methodology. Modeling of power engineering processes using SDE enables consideration of the risk in simulation.
7
Content available Solution of linear and nonlinear diffusion problems via stochastic differential equations
Bargieł M., Tory E. M.
Computer Science
|
2015
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Vol. 16 (4)
415--428
EN
The equation for nonlinear diffusion can be rearranged to a form that immediately leads to its stochastic analog. The latter contains a drift term that is absent when the diffusion coefficient is constant. The dependence of this coefficient on concentration (or temperature) is handled by generating many paths in parallel and approximating the derivative of concentration with respect to distance by the central difference. This method works for one-dimensional diffusion problems with finite or infinite boundaries and for diffusion in cylindrical or spherical shells. By mimicking the movements of molecules, the stochastic approach provides a deeper insight into the physical process. The parallel version of our algorithm is very efficient. The 99% confidence limits for the stochastic solution enclose the analytical solution so tightly that they cannot be shown graphically. This indicates that there is no systematic difference in the results for the two methods. Finally, we present a direct derivation of the stochastic method for cylindrical and spherical shells.
8
Model typu „end-use” prognozy zapotrzebowania na energię pierwotną i finalną w warunkach ryzyka
Sowiński J.
Rynek Energii
|
2014
|
Nr 3
39--43
PL
Rynkowy charakter gospodarki energetycznej powoduje, że w okresach zmian strukturalnych i kryzysowych modele globalne wymagają modyfikacji metodyk, gdyż załamują się dotychczasowe trendy, a konieczne jest uwzględnienie niepewności otoczenia oraz ryzyka finansowego, inwestycyjnego itp. Wskaźniki energochłonności umożliwiają monitoro¬wanie gospodarki energetycznej kraju, dając podstawę do realizacji predykcji zapotrzebowania na energię pierwotną i fi¬nalną modelami typu „end-use”. Przebiegi współczynników energochłonności można zamodelować stochastycznymi równa¬niami różniczkowymi. Rozwiązanie równań metodą Eulera umożliwia symulację przebiegów w horyzoncie średniotermino¬wym. Na tej podstawie zbudowano model typu „end-use” prognozy zapotrzebowania na energię, który może wspierać analizę bilansu energetycznego kraju. Przedstawiono przykładowe wyniki symulacji, które powinny być weryfikowane ocenami eksperckimi.
EN
The market rules of energy management require deep improvement of global models methodology. The historical drifts have collapsed. The uncertainty of energy processes cause financial and investment risk. The control of energy efficiency coefficients enables monitoring of national energy economy. The coefficients are necessary in forecasting of primary and final energy demand in the “end-use” models. Stochastic differential equations characterize the behavior of energy efficiency coefficients as continuous time stochastic processes. Numerical solution of stochastic differential equations could use the Euler method and enables simulation of stochastic variables in the medium-term horizon. The end-use model of energy forecast has been constructed. The model may support research on energy balance. The simulation results, which should be verified using expert estimations, are presented.
9
Content available remote Zastosowanie pakietu Matlab do rozwiązywania stochastycznych równań różniczkowych
Mazurkiewicz S., Walczak J.
Prace Naukowe Politechniki Śląskiej. Elektryka
|
2013
|
z. 2-3
7--14
PL
W artykule omówiono możliwości pakietu SDE Toolbox przeznaczonego do rozwiązywania stochastycznych równań różniczkowych. Przedstawiono podstawowe wady i zalety założeń przyjętych przez autora pakietu. Pokazano podstawy obsługi pakietu.
EN
The article discusses the possibility of the SDE Toolbox intended for solving stochastic differential equations. The basic advantages and disadvantages of the assumptions made by the author of the SDE Toolbox are shown. There are shown usage of the toolbox.
10
Content available remote Lower bound technique and its applications to function systems and stochastic partial differential equations
Szarek T.
Mathematica Applicanda
|
2013
|
Vol. 41, No. 2
185--198
EN
We formulate some criteria for the existence of an invariant measure for Markov chains and Markov processes. We also show their application in the theory of function systems and stochastic differential equations
PL
W pracy formułujemy kryteria dla istnienia miary niezmienniczej dla łańcuchów i procesów Markowa. Następnie pokazujemy ich użyteczność w teorii iterowanych układów funkcyjnych i stochastycznych równań rózniczkowych.
11
Content available remote Generalized backward doubly stochastic differential equations driven by Lévy processes with non-Lipschitz coefficients
Aman A., Owo J.-M.
Probability and Mathematical Statistics
|
2010
|
Vol. 30, Fasc. 2
259--272
EN
We prove an existence and uniqueness result for generalized backward doubly stochastic differential equations driven by Lévy processes with non-Lipschitz assumptions.
12
Content available remote Discrete approximations of reflected backward stochastic differential equations with random terminal time
Jańczak K.
Probability and Mathematical Statistics
|
2008
|
Vol. 28, Fasc. 1
41--74
EN
We study convergence of discrete approximations of reflected backward stochastic differential equations with random terminal time in a general convex domain. Applications to investigation of the viability property for backward stochastic differential equations and to obstacle problem for partial differential equations are given.
13
Content available remote The probabilistic solution of the Dirichlet problem for degenerate elliptic equations
Liu C., Morimoto H.
Bulletin of the Polish Academy of Sciences. Mathematics
|
2006
|
Vol. 54, no 2
163-174
EN
We study the Dirichlet problem for degenerate elliptic equations, and show that the probabilistic solution is a unique viscosity solution.
14
Content available remote Noise simulation of linear active circuits by numerical solution of stochastic differential equations
Beute B., Mathis W., Markovic V.
Przegląd Elektrotechniczny
|
2003
|
R. 79, nr 10
779-782
EN
In this paper a concept for simulating noise in linear active circuits in time domain is presented. For this purpose corresponding stochastic differential equations are formulated and solved by stochastic integration methods. Some circuit examples including thermal noise illustrate our approach.
PL
Praca przedstawia nową koncepcję symulacji szumu obwodów liniowych aktywnych w dziedzinie czasu. W tym celu formułuje się równania różniczkowe stochastyczne, rozwiązywane metodami numerycznymi charakterystycznymi dla tego typu równań. Przykłady obwodów z szumami termalnymi analizowanych tymi metodami zamieszczono również w pracy.
15
Content available remote Stochastic evolutions driven by non-linear white noise
Accardi L., Boukas A.
Probability and Mathematical Statistics
|
2002
|
Vol. 22, Fasc. 1
141--154
EN
We prove the existence and uniqueness theorem for stochastic differential equations with bounded coefficients driven by the renormalized square of white noise.These equations are interpreted as sesquilinear forms on the linear span of the exponential vectors (of the first order white noise) and the existence theorem is establishedon the space of these forms.
16
Content available remote On invariance of linear subspaces for stochastic evolution equations
Milian A.
Demonstratio Mathematica
|
2001
|
Vol. 34, nr 2
425-430
EN
In the paper we consider Ito equation on a Hilbert space. We give necessary and sufficient conditions ensuring the invariance property of linear subspaces of the state space by mild solutions to the equation.
17
Content available remote A closure method for randomly perturbed linear systems
Bobryk R., Stettener Ł.
Demonstratio Mathematica
|
2001
|
Vol. 34, nr 2
415-425
18
Content available remote Cellular Automata and Many-Particle Systems Modeling Aggregation Behavior Among Populations
Morale D.
International Journal of Applied Mathematics and Computer Science
|
2000
|
Vol. 10, no 1
157-173
EN
A cellular automaton model is presented in order to describe mutual interactions among the individuals of a population due to social decisions.The scheme is used for getting qualitative results, comparable to field experiments carried out on a population of ants which present an aggregative behavior. We also present a second description of a biological spatially structured population of N individuals by a system of stochastic differential equations of Ito type. A 'law of large numbers' to a continuum dynamics described by an integro-differential equation is given.
19
Content available Application of the votex-in-cell method for the simulation of two-dimensional viscous flow
Kundela H.
TASK Quarterly : scientific bulletin of Academic Computer Centre in Gdansk
|
1999
|
Vol. 3, No 3
343-360
EN
In the paper the vortex in cell method for the simulation of the viscous flow in a complex geometry was described. Vorticity field is approximated by the collection of the particles that carries the circulation. The local velocity of a particle was obtained by the solution of the Poisson equation for the stream function by the grid method and then interpolation of velocity from the grid nodes to the vortex particle position. The Poisson equation for the stream function was solved by fast elliptic solvers. To be able to solve the Poisson equation in a region with a complex geometry, the capacitance matrix technique was used. The viscosity of the fluid was taken in a stochastic manner. A suitable stochastic differential equation was solved by the Huen method. The non-slip condition on the wall was realized by the generation of the vorticity. The program was tested by solving several flows in the channels with a different geometry and at a different Reynolds number. Here we present the testing results concerning the flow in a channel with sudden symmetric expansion, for the flow in channel with backward step, and the flow over building systems.
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