17 results on '"Wiener process"'
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2. The First-Passage Area of Wiener Process with Stochastic Resetting
- Author
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Abundo, Mario
- Published
- 2023
- Full Text
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3. Mean-square convergence analysis of the semi-implicit scheme for stochastic differential equations driven by the Wiener processes
- Author
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Torkzadeh, L.
- Published
- 2023
- Full Text
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4. Measure of noncompactness and application to stochastic differential equations.
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Dehici, Abdelkader and Redjel, Nadjeh
- Subjects
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COMPACT spaces (Topology) , *STOCHASTIC differential equations , *RANDOM operators , *EXISTENCE theorems , *FIXED point theory - Abstract
In this paper, we study the existence and uniqueness of the solution of stochastic differential equation by means of the properties of the associated condensing nonexpansive random operator. Moreover, by taking account of the results of Diaz and Metcalf, we prove the convergence of Kirk's process to this solution for small times. [ABSTRACT FROM AUTHOR]
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- 2016
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5. Stochastic accessibility on Grushin-type manifolds.
- Author
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Ţurcanu, Teodor and Udrişte, Constantin
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NONSMOOTH optimization , *DISTRIBUTION (Probability theory) , *STOCHASTIC processes , *MANIFOLDS (Mathematics) , *ASYMPTOTIC efficiencies - Abstract
We consider a non-smooth Grushin-type distribution, defined on R n , whose stochastic perturbation defines the admissible stochastic processes. Our main result is a stochastic accessibility theorem on the corresponding Grushin manifold. More specifically, given two points P and Q , we show how to steer an admissible stochastic processes, starting at P , such that it strikes an arbitrarily small ball centered at Q , asymptotically almost surely. [ABSTRACT FROM AUTHOR]
- Published
- 2017
- Full Text
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6. Existence and exponential stability for some stochastic neutral partial functional integrodifferential equations.
- Author
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Diop, Mamadou Abdou, Ezzinbi, Khalil, and Lo, Modou
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INTEGRO-differential equations , *EXISTENCE theorems , *EXPONENTIAL stability , *FUNCTIONAL equations , *STOCHASTIC differential equations , *ITERATIVE methods (Mathematics) - Abstract
The aim of this work is to study the existence, uniqueness and exponential stability of mild solutions for some stochastic neutral partial functional integrodifferential equations. We suppose that the linear part has a resolvent operator in the sense given in Grimmer [Transactions of the American Mathematical Society 273 (1982), 333-349]. The nonlinear part is assumed to be continuous and lipschitzian with respect to the second argument. Firstly, we study the existence of mild solutions. Secondly we give some results on the exponential stability in mean square sense. An example is provided to illustrate the results of this work. [ABSTRACT FROM AUTHOR]
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- 2014
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- View/download PDF
7. Mixed fractional stochastic differential equations with jumps.
- Author
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Shevchenko, Georgiy
- Subjects
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STOCHASTIC differential equations , *WIENER processes , *MOMENTS method (Statistics) , *POISSON processes , *PROBABILITY theory , *STOCHASTIC processes , *MATHEMATICAL analysis - Abstract
In this paper, we consider a stochastic differential equation driven by a fractional Brownian motion and a Wiener process and having jumps. We prove that this equation has a unique solution and show that all moments of the solution are finite. [ABSTRACT FROM AUTHOR]
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- 2014
- Full Text
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8. Malliavin regularity of solutions to mixed stochastic differential equations.
- Author
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Shevchenko, Georgiy and Shalaiko, Taras
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MALLIAVIN calculus , *MATHEMATICAL regularization , *NUMERICAL solutions to stochastic differential equations , *BROWNIAN motion , *WIENER processes , *EXISTENCE theorems , *MATHEMATICAL proofs - Abstract
Abstract: For a mixed stochastic differential equation driven by independent fractional Brownian motions and Wiener processes, the existence and integrability of the Malliavin derivative of the solution are established. It is also proved that the solution possesses exponential moments. [Copyright &y& Elsevier]
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- 2013
- Full Text
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9. Rate of convergence and asymptotic error distribution of Euler approximation schemes for fractional diffusions
- Author
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David Nualart, Yaozhong Hu, and Yanghui Liu
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Statistics and Probability ,Euler scheme ,Malliavin calculus ,fractional calculus ,01 natural sciences ,Fractional Brownian motion ,Euler method ,010104 statistics & probability ,symbols.namesake ,Stochastic differential equation ,60H07 ,Wiener process ,Mathematics::Probability ,FOS: Mathematics ,0101 mathematics ,fourth moment theorem ,Mathematical physics ,Mathematics ,Weak convergence ,010102 general mathematics ,Probability (math.PR) ,16. Peace & justice ,stochastic differential equations ,Distribution (mathematics) ,Rate of convergence ,Euler's formula ,symbols ,60H35 ,60H10 ,Statistics, Probability and Uncertainty ,26A33 ,Mathematics - Probability - Abstract
For a stochastic differential equation(SDE) driven by a fractional Brownian motion(fBm) with Hurst parameter $H>\frac{1}{2}$, it is known that the existing (naive) Euler scheme has the rate of convergence $n^{1-2H}$. Since the limit $H\rightarrow\frac{1}{2}$ of the SDE corresponds to a Stratonovich SDE driven by standard Brownian motion, and the naive Euler scheme is the extension of the classical Euler scheme for It\^{o} SDEs for $H=\frac{1}{2}$, the convergence rate of the naive Euler scheme deteriorates for $H\rightarrow\frac{1}{2}$. In this paper we introduce a new (modified Euler) approximation scheme which is closer to the classical Euler scheme for Stratonovich SDEs for $H=\frac{1}{2}$, and it has the rate of convergence $\gamma_n^{-1}$, where $\gamma_n=n^{2H-{1}/2}$ when $H\frac{3}{4}$. Furthermore, we study the asymptotic behavior of the fluctuations of the error. More precisely, if $\{X_t,0\le t\le T\}$ is the solution of a SDE driven by a fBm and if $\{X_t^n,0\le t\le T\}$ is its approximation obtained by the new modified Euler scheme, then we prove that $\gamma_n(X^n-X)$ converges stably to the solution of a linear SDE driven by a matrix-valued Brownian motion, when $H\in(\frac{1}{2},\frac{3}{4}]$. In the case $H>\frac{3}{4}$, we show the $L^p$ convergence of $n(X^n_t-X_t)$, and the limiting process is identified as the solution of a linear SDE driven by a matrix-valued Rosenblatt process. The rate of weak convergence is also deduced for this scheme. We also apply our approach to the naive Euler scheme., Comment: Published at http://dx.doi.org/10.1214/15-AAP1114 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)
- Published
- 2016
10. Gaussian asymptotics for a non-linear Langevin type equation driven by an $\alpha$-stable Lévy noise
- Author
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Richard Eon, Mihai Gradinaru, Institut de Recherche Mathématique de Rennes ( IRMAR ), Université de Rennes 1 ( UR1 ), Université de Rennes ( UNIV-RENNES ) -Université de Rennes ( UNIV-RENNES ) -AGROCAMPUS OUEST-École normale supérieure - Rennes ( ENS Rennes ) -Institut National de Recherche en Informatique et en Automatique ( Inria ) -Institut National des Sciences Appliquées ( INSA ) -Université de Rennes 2 ( UR2 ), Université de Rennes ( UNIV-RENNES ) -Centre National de la Recherche Scientifique ( CNRS ), Institut de Recherche Mathématique de Rennes (IRMAR), Université de Rennes (UR)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-École normale supérieure - Rennes (ENS Rennes)-Université de Rennes 2 (UR2)-Centre National de la Recherche Scientifique (CNRS)-INSTITUT AGRO Agrocampus Ouest, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro), AGROCAMPUS OUEST, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Université de Rennes 1 (UR1), Université de Rennes (UNIV-RENNES)-Université de Rennes (UNIV-RENNES)-Université de Rennes 2 (UR2), Université de Rennes (UNIV-RENNES)-École normale supérieure - Rennes (ENS Rennes)-Centre National de la Recherche Scientifique (CNRS)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), and Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA)
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Statistics and Probability ,[ MATH ] Mathematics [math] ,01 natural sciences ,Lévy process ,non-linear Langevin type equation ,010104 statistics & probability ,Stochastic differential equation ,symbols.namesake ,stable Lévy noise ,Wiener process ,60J65 ,60G44 ,[MATH]Mathematics [math] ,0101 mathematics ,Power function ,exponential ergodic processes ,Brownian motion ,Mathematics ,Lévy driven stochastic dierential equa-tion ,Lyapunov function ,010102 general mathematics ,Mathematical analysis ,Lévy driven stochastic differential equation ,[MATH.MATH-PR]Mathematics [math]/Probability [math.PR] ,Scaling limit ,Convergence of random variables ,Diffusion process ,functional central limit theorem for martingales ,60F17 ,60G52 ,60J75 ,60H10 ,convergence in probability ,symbols ,Statistics, Probability and Uncertainty ,[ MATH.MATH-PR ] Mathematics [math]/Probability [math.PR] - Abstract
International audience; Consider a one-dimensional process $x^{\varepsilon}_{t}$ the position of a particle at time $t$ which speed $v^{\varepsilon}_{t}$ is a solution of a stochastic differential equation driven by a small $\alpha$-stable Lévy process, $\varepsilon\ell_t$, $\alpha\in(0,2]$, and with a non-linear drift coefficient $-{\rm sgn}(v)|v|^{\beta}$, $\beta>2-(\frac{\alpha}{2})$. The noise could be path continuous (Brownian motion $\alpha=2$) or pure jump process ($0
- Published
- 2015
11. Multilevel Monte Carlo algorithms for L\'{e}vy-driven SDEs with Gaussian correction
- Author
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Steffen Dereich
- Subjects
Statistics and Probability ,weak approximation ,Gaussian ,Monte Carlo method ,Komlós–Major–Tusnády coupling ,Lipschitz continuity ,Lévy process ,Upper and lower bounds ,Lévy-driven stochastic differential equation ,Stochastic differential equation ,symbols.namesake ,Uniform norm ,Mathematics::Probability ,Wiener process ,60H05 ,Multilevel Monte Carlo ,symbols ,numerical integration ,60H35 ,60H10 ,Statistics, Probability and Uncertainty ,60J75 ,Algorithm ,Mathematics - Probability ,Mathematics - Abstract
We introduce and analyze multilevel Monte Carlo algorithms for the computation of $\mathbb {E}f(Y)$, where $Y=(Y_t)_{t\in[0,1]}$ is the solution of a multidimensional L\'{e}vy-driven stochastic differential equation and $f$ is a real-valued function on the path space. The algorithm relies on approximations obtained by simulating large jumps of the L\'{e}vy process individually and applying a Gaussian approximation for the small jump part. Upper bounds are provided for the worst case error over the class of all measurable real functions $f$ that are Lipschitz continuous with respect to the supremum norm. These upper bounds are easily tractable once one knows the behavior of the L\'{e}vy measure around zero. In particular, one can derive upper bounds from the Blumenthal--Getoor index of the L\'{e}vy process. In the case where the Blumenthal--Getoor index is larger than one, this approach is superior to algorithms that do not apply a Gaussian approximation. If the L\'{e}vy process does not incorporate a Wiener process or if the Blumenthal--Getoor index $\beta$ is larger than $\frac{4}{3}$, then the upper bound is of order $\tau^{-({4-\beta})/({6\beta})}$ when the runtime $\tau$ tends to infinity. Whereas in the case, where $\beta$ is in $[1,\frac{4}{3}]$ and the L\'{e}vy process has a Gaussian component, we obtain bounds of order $\tau^{-\beta/(6\beta-4)}$. In particular, the error is at most of order $\tau^{-1/6}$., Comment: Published in at http://dx.doi.org/10.1214/10-AAP695 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)
- Published
- 2011
12. $p$-th Mean Pseudo Almost Automorphic Mild Solutions to Some Nonautonomous Stochastic Differential Equations
- Author
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Bezandry, P. H. and Diagana, T.
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Wiener process ,34K14 ,Stochastic differential equation ,square-mean pseudo almost automorphic ,square-mean pseudo almost periodic ,34F05 ,60H10 ,$p$-th mean pseudo almost automorphic ,$p$-th mean pseudo almost periodic ,35B15 - Abstract
In this paper we first introduce and study the concepts of $p$-th mean pseudo almost automorphy and that of $p$-th mean pseudo almost periodicity for $p \geq 2$. Next, we make extensive use of the well-known Schauder fixed point principle to obtain the existence of $p$-th mean pseudo almost automorphic mild solutions to some nonautonomous stochastic differential equations.
- Published
- 2011
13. Solutions of Stochastic Differential Equations obeying the Law of the Iterated Logarithm, with applications to financial markets
- Author
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Huizhong Wu and John A. D. Appleby
- Subjects
Statistics and Probability ,Geometric Brownian motion ,Class (set theory) ,Mathematical analysis ,Law of the iterated logarithm ,inefficient market ,Space (mathematics) ,stochastic differential equations ,91B28 ,Stochastic partial differential equation ,Stochastic differential equation ,symbols.namesake ,stochastic comparison principle ,Wiener process ,symbols ,stationary processes ,Applied mathematics ,Law of the Iterated Logarithm ,60H10 ,Statistics, Probability and Uncertainty ,Brownian motion ,Motoo's theorem ,Mathematics ,60F10 - Abstract
By using a change of scale and space, we study a class of stochastic differential equations (SDEs) whose solutions are drift--perturbed and exhibit asymptotic behaviour similar to standard Brownian motion. In particular sufficient conditions ensuring that these processes obey the Law of the Iterated Logarithm (LIL) are given. Ergodic--type theorems on the average growth of these non-stationary processes, which also depend on the asymptotic behaviour of the drift coefficient, are investigated. We apply these results to inefficient financial market models. The techniques extend to certain classes of finite--dimensional equation.
- Published
- 2009
14. Exponential Asymptotic Stability of Linear Itô-Volterra Equation with Damped Stochastic Perturbations
- Author
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Alan Freeman and John A. D. Appleby
- Subjects
Statistics and Probability ,Discrete mathematics ,Zero (complex analysis) ,34K20 ,Square (algebra) ,Exponential function ,Combinatorics ,Kernel (algebra) ,symbols.namesake ,Exponential stability ,Rate of convergence ,Wiener process ,symbols ,60H10 ,Statistics, Probability and Uncertainty ,45D05 ,60H20 ,Sign (mathematics) ,Mathematics - Abstract
This paper studies the convergence rate of solutions of the linear Itô-Volterra equation $$ dX(t) = \left(AX(t) + \int_{0}^{t} K(t-s)X(s),ds\right)\,dt + \Sigma(t)\,dW(t) \tag{1} $$ where $K$ and $\Sigma$ are continuous matrix-valued functions defined on $\mathbb{R}^{+}$, and $(W(t))_{t \geq 0}$ is a finite-dimensional standard Brownian motion. It is shown that when the entries of $K$ are all of one sign on $\mathbb{R}^{+}$, that (i) the almost sure exponential convergence of the solution to zero, (ii) the $p$-th mean exponential convergence of the solution to zero (for all $p \gt 0$), and (iii) the exponential integrability of the entries of the kernel $K$, the exponential square integrability of the entries of noise term $\Sigma$, and the uniform asymptotic stability of the solutions of the deterministic version of (1) are equivalent. The paper extends a result of Murakami which relates to the deterministic version of this problem.
- Published
- 2003
15. Levy area of Wiener processes in Banach spaces
- Author
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M. Ledoux, Zhongmin Qian, and Terry Lyons
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Statistics and Probability ,Pure mathematics ,Rough path ,Gaussian comparison theorem ,rough path ,Stochastic process ,Mathematical analysis ,Banach space ,Gaussian measure ,Lévy process ,differential equation ,symbols.namesake ,Stochastic differential equation ,Mathematics::Probability ,Wiener process ,60G15 ,60H15 ,symbols ,60H10 ,Tensor ,Brownian motion ,Statistics, Probability and Uncertainty ,60J60 ,Mathematics - Abstract
The goal of this paper is to construct canonical Levy area processes for Banach space valued Brownian motions via dyadic approximations. The significance of the existence of canonical Levy area processes is that a (stochastic) integration theory can be established for such Brownian motions (in Banach spaces). Existence of flows for stochastic differential equations with infinite dimensional noise then follows via the results of Lyons and Lyons and Qian [see, e.g., System Control and Rough Paths (2000). Oxford Univ. Press]. This investigation involves a careful analysis on the choice of tensor norms, motivated by the applications to infinite dimensional stochastic differential equations.
- Published
- 2002
16. Variably Skewed Brownian Motion
- Author
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Martin T. Barlow, Avi Mandelbaum, Haya Kaspi, and Krzysztof Burdzy
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Statistics and Probability ,Geometric Brownian motion ,Fractional Brownian motion ,Mathematical analysis ,Brownian excursion ,stochastic differential equation ,symbols.namesake ,Diffusion process ,Wiener process ,Reflected Brownian motion ,Mathematics::Probability ,local time ,symbols ,60J65 ,Differentiable function ,60H10 ,Statistics, Probability and Uncertainty ,Skew Brownian motion ,Brownian motion ,Mathematics ,Mathematical physics - Abstract
Given a standard Brownian motion $B$, we show that the equation $$ X_t = x_0 + B_t + \beta(L_t^X), t \geq 0,$$ has a unique strong solution $X$. Here $L^X$ is the symmetric local time of $X$ at $0$, and $\beta$ is a given differentiable function with $\beta(0) = 0$, whose derivative is always in $(-1,1)$. For a linear function $\beta$, the solution is the familiar skew Brownian motion.
- Published
- 2000
17. Large Deviations for a Class of Anticipating Stochastic Differential Equations
- Author
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Annie Millet, Marta Sanz, and David Nualart
- Subjects
Statistics and Probability ,Stratonovich integral ,anticipating stochastic differential equations ,Mathematical analysis ,Stochastic calculus ,Malliavin calculus ,Stochastic partial differential equation ,Combinatorics ,Stochastic differential equation ,symbols.namesake ,Large deviations ,Quantum stochastic calculus ,Wiener process ,stochastic flows ,symbols ,Large deviations theory ,60H10 ,Statistics, Probability and Uncertainty ,Mathematics ,60F10 - Abstract
Consider the family of perturbed stochastic differential equations on $\mathbb{R}^d$, $X^\varepsilon_t = X^\varepsilon_0 + \sqrt{\varepsilon} \int^t_0\sigma(X^\varepsilon_s)\circ dW_s + \int^t_0 b(X^\varepsilon_s) ds,$ $\varepsilon > 0$, defined on the canonical space associated with the standard $k$-dimensional Wiener process $W$. We assume that $\{X^\varepsilon_0, \varepsilon > 0\}$ is a family of random vectors not necessarily adapted and that the stochastic integral is a generalized Stratonovich integral. In this paper we prove large deviations estimates for the laws of $\{X^\varepsilon_., \varepsilon > 0\}$, under some hypotheses on the family of initial conditions $\{X^\varepsilon_0, \varepsilon > 0\}$.
- Published
- 1992
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