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1
Content available remote Stochastic evolutions driven by non-linear white noise
100%
Accardi L. , Boukas A.
Probability and Mathematical Statistics
|
2002
|
tom 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.
2
Content available remote Zastosowanie pakietu Matlab do rozwiązywania stochastycznych równań różniczkowych
100%
Mazurkiewicz S. , Walczak J.
Prace Naukowe Politechniki Śląskiej. Elektryka
|
2013
|
tom 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.
3
Content available remote Discrete approximations of reflected backward stochastic differential equations with random terminal time
100%
Jańczak K.
Probability and Mathematical Statistics
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2008
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tom 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.
4
Content available remote Robust Control of Linear Stochastic Systems with Fully Observable State
100%
Poznyak A. S. , Taksar M. I.
Applicationes Mathematicae
|
1996-1997
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tom 24
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nr 1
35-46
EN
We consider a multidimensional linear system with additive inputs (control) and Brownian noise. There is a cost associated with each control. The aim is to minimize the cost. However, we work with the model in which the parameters of the system may change in time and in addition the exact form of these parameters is not known, only intervals within which they vary are given. In the situation where minimization of a functional over the class of admissible controls makes no sense since the value of such a functional is different for different systems within the class, we should deal not with a single problem but with a family of problems. The objective in such a setting is twofold. First, we intend to establish existence of a state feedback linear robust control which stabilizes any system within the class. Then among all robust controls we find the one which yields the lowest bound on the cost within the class of all systems under consideration. We give the answer in terms of a solution to a matrix Riccati equation and we present necessary and sufficient conditions for such a solution to exist. We also state a criterion when the obtained bound on the cost is sharp, that is, the control we construct is actually a solution to the minimax problem.
5
Content available remote A Note on Existence of Global Solutions and Invariant Measures for Jump Sdes with Locally One-Sided Lipschitz Drift
100%
Majka M. B.
Probability and Mathematical Statistics
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2020
|
tom 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.
6
Content available remote Weak convergence of a numerical scheme for stochastic differential equations
100%
Aguilera E. , Fierro R.
Probability and Mathematical Statistics
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2017
|
tom 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.
7
Content available remote On invariance of linear subspaces for stochastic evolution equations
100%
Milian A.
Demonstratio Mathematica
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2001
|
tom 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.
8
Content available remote A closure method for randomly perturbed linear systems
100%
Bobryk R. , Stettener Ł.
Demonstratio Mathematica
|
2001
|
tom Vol. 34, nr 2
415-425
9
Content available remote The probabilistic solution of the Dirichlet problem for degenerate elliptic equations
100%
Liu C. , Morimoto H.
Bulletin of the Polish Academy of Sciences. Mathematics
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2006
|
tom 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.
10
Content available remote Generalized backward doubly stochastic differential equations driven by Lévy processes with non-Lipschitz coefficients
88%
Aman A. , Owo J.-M.
Probability and Mathematical Statistics
|
2010
|
tom 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.
11
Content available remote 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.
12
Content available Review of stochastic differential equations in statistical arbitrage pairs trading
88%
Endres S.
Managerial Economics
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2019
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tom 20
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nr 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 meanreversion, 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 fur future research are indicated.
13
Content available Measure valued solutions for stochastic evolution equations on Hilbert space and their feedback control
88%
Ahmed N.
Discussiones Mathematicae, Differential Inclusions, Control and Optimization
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2005
|
tom 25
|
nr 1
129-157
EN
In this paper, we consider a class of semilinear stochastic evolution equations on Hilbert space driven by a stochastic vector measure. The nonlinear terms are assumed to be merely continuous and bounded on bounded sets. We prove the existence of measure valued solutions generalizing some earlier results of the author. As a corollary, an existence result of a measure solution for a forward Kolmogorov equation with unbounded operator valued coefficients is obtained. The main result is further extended to cover Borel measurable drift and diffusion which are assumed to be bounded on bounded sets. Also we consider control problems for these systems and present several results on the existence of optimal feedback controls.
14
Content available Solution of linear and nonlinear diffusion problems via stochastic differential equations
88%
Bargieł M. , Tory E. M.
Computer Science
|
2015
|
tom 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.
15
Content available remote Noise simulation of linear active circuits by numerical solution of stochastic differential equations
75%
Beute B. , Mathis W. , Markovic V.
Przegląd Elektrotechniczny
|
2003
|
tom 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.
16
Content available O aproksymacjach funkcji przejścia dla jednowymiarowych procesów dyfuzji
75%
Szczepocki P.
Przegląd Statystyczny
|
2016
|
tom 63
|
nr 2
173-190
EN
Estimation methods for stochastic differentia equations driver by discretely sampled continuous diffusion processes may be split into two categories: maximum likelihood methods and methods based on the general method of moments. Usually, one does not know neither likelihood function nor theoretical moments of diffusion process and cannot construct estimators. Therefore many methods was developed to approximating unknown transition density. The aim of article is to compare properties of selected approaches, indicate their merits and limitations.
PL
Metody estymacji parametrów stochastycznych równań różniczkowych dla ciągłych procesów dyfuzji obserwowanych w dyskretnych odstępach czasu można podzielić na dwie kategorie: metody oparte na maksymalizacji funkcji wiarygodności i wykorzystujące uogólnioną metodę momentów. Zazwyczaj nie znamy jednak gęstości przejścia potrzebnej do konstrukcji funkcji wiarygodności, ani odpowiedniej ilości momentów teoretycznych, aby skonstruować odpowiednią liczbę warunków. Dlatego powstało wiele metod, które próbują przybliżyć nieznaną funkcję przejścia. Celem artykułu jest porównanie własności wybranych metod aproksymacji jednowymiarowych jednorodnych procesów dyfuzji.
17
Content available remote Cross-variation of Young integral with respect to long-memory fractional Brownian motions
75%
Nourdin I. , Zintout R.
Probability and Mathematical Statistics
|
2016
|
tom 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.
18
Content available Stochastic diffrential equations on Banach spaces and their optimal feedback control
75%
Discussiones Mathematicae, Differential Inclusions, Control and Optimization
|
2012
|
tom 32
|
nr 1
87-109
EN
In this paper we consider stochastic differential equations on Banach spaces (not Hilbert). The system is semilinear and the principal operator generating a C₀-semigroup is perturbed by a class of bounded linear operators considered as feedback operators from an admissible set. We consider the corresponding family of measure valued functions and present sufficient conditions for weak compactness. Then we consider applications of this result to several interesting optimal feedback control problems. We present results on existence of optimal feedback operators.
19
Content available Stochastic models of the slow/fast type of atrioventricular nodal reentrant tachycardia and tachycardia with conduction aberration
63%
Jackowska-Zduniak B.
International Journal of Applied Mathematics and Computer Science
|
2022
|
tom 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.
20
Content available remote Lower bound technique and its applications to function systems and stochastic partial differential equations
63%
Szarek T.
Mathematica Applicanda
|
2013
|
tom 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.
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