Author
Bilel Kacem Ben Ammou
Bio: Bilel Kacem Ben Ammou is an academic researcher from Tunis University. The author has contributed to research in topics: Stochastic differential equation & Brownian motion. The author has an hindex of 2, co-authored 5 publications receiving 7 citations.
Papers
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TL;DR: In this article, the authors consider a class of stochastic differential equations driven by a one-dimensional Brownian motion, and investigate the rate of convergence for Wong-Zakai-type approximated solutions.
Abstract: We consider a class of stochastic differential equations driven by a one-dimensional Brownian motion, and we investigate the rate of convergence for Wong–Zakai-type approximated solutions. We first consider the Stratonovich case, obtained through the pointwise multiplication between the diffusion coefficient and a smoothed version of the noise; then, we consider Ito equations where the diffusion coefficient is Wick-multiplied by the regularized noise. We discover that in both cases the speed of convergence to the exact solution coincides with the speed of convergence of the smoothed noise toward the original Brownian motion. We also prove, in analogy with a well-known property for exact solutions, that the solutions of approximated Ito equations solve approximated Stratonovich equations with a certain correction term in the drift.
6 citations
TL;DR: In this paper, the authors consider a class of stochastic differential equations driven by a one dimensional Brownian motion and investigate the rate of convergence for Wong-Zakai-type approximated solutions.
Abstract: We consider a class of stochastic differential equations driven by a one dimensional Brownian motion and we investigate the rate of convergence for Wong-Zakai-type approximated solutions. We first consider the Stratonovich case, obtained through the point-wise multiplication between the diffusion coefficient and a smoothed version of the noise; then, we consider Ito equations where the diffusion coefficient is Wick-multiplied by the regularized noise. We discover that in both cases the speed of convergence to the exact solution coincides with the speed of convergence of the smoothed noise towards the original Brownian motion. We also prove, in analogy with a well known property for exact solutions, that the solutions of approximated Ito equations solve approximated Stratonovich equations with a certain correction term in the drift.
4 citations
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TL;DR: In this article, the authors investigated the properties of the Wick square of Gaussian white noises through a new method to perform non linear operations on Hida distributions, which lays in between the Wick product interpretation and the usual definition of nonlinear functions.
Abstract: We investigate the properties of the Wick square of Gaussian white noises through a new method to perform non linear operations on Hida distributions. This method lays in between the Wick product interpretation and the usual definition of nonlinear functions. We prove on Ito-type formula and solve stochastic differential equations driven by the renormalized square of the Gaussian white noise. Our approach works with standard assumptions on the coefficients of the equations, Lipschitz continuity and linear growth condition, and produces existence and uniqueness results in the space where the noise lives. The linear case is studied in details and positivity of the solution is proved.
TL;DR: In this paper, the authors investigated the properties of the Wick square of Gaussian white noises through a new method to perform nonlinear operations on Hida distributions, which lays in between the Wick product interpretation and the usual definition of nonlinear functions.
Abstract: We investigate the properties of the Wick square of Gaussian white noises through a new method to perform nonlinear operations on Hida distributions. This method lays in between the Wick product interpretation and the usual definition of nonlinear functions. We prove an Ito-type formula and solve stochastic differential equations driven by the renormalized square of the Gaussian white noise. Our approach works with standard assumptions on the coefficients of the equations, global Lipschitz continuity, and produces existence and uniqueness results in the space where the noise lives. The linear case is studied in details and positivity of the solution is proved.
TL;DR: By means of infinite-dimensional nuclear spaces, the authors generalize important results on the representation of the Weyl commutation relations, and construct a new nuclear Lie group gene gene.
Abstract: By means of infinite-dimensional nuclear spaces, we generalize important results on the representation of the Weyl commutation relations. For this purpose, we construct a new nuclear Lie group gene...
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TL;DR: In this article, for any finite partition of the time interval [ 0, T ] a system of differential equations, where the multidimensional Brownian motion is replaced by its polygonal approximation and the product between diffusion coefficients and smoothed white noise is interpreted as a Wick product, is considered, with the aim of approximating the solution of a quasilinear system of Ito SDEs.
4 citations
TL;DR: In this paper, the authors investigate the regularity of the law of Wong-Zakai-type approximations for Ito stochastic differential equations, and establish absolute continuity and a Fokker-Planck-type equation solved in the distributional sense by the density.
3 citations
TL;DR: In this article , Wong-Zakai approximation results for two-and three-dimensional stochastic convective Brinkman-Forchheimer (SCBF) equations forced by Hilbert space valued Wiener noise on bounded domains were demonstrated.
1 citations
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TL;DR: In this article, the authors investigate the regularity of the law of Wong-Zakai-type approximations for Ito stochastic differential equations, and establish absolute continuity and a Fokker-Planck-type equation solved in the distributional sense by the density.
Abstract: We investigate the regularity of the law of Wong-Zakai-type approximations for Ito stochastic differential equations. These approximations solve random differential equations where the diffusion coefficient is Wick-multiplied by the smoothed white noise. Using a criteria based on the Malliavin calculus we establish absolute continuity and a Fokker-Planck-type equation solved in the distributional sense by the density. The parabolic smoothing effect typical of the solutions of Ito equations is lacking in this approximated framework; therefore, in order to prove absolute continuity, the initial condition of the random differential equation needs to possess a density itself.
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TL;DR: In this article, for any finite partition of the time interval $[0,T]$ a system of differential equations, where the multidimensional Brownian motion is replaced by its polygonal approximation and the product between diffusion coefficients and smoothed white noise is interpreted as a Wick product, is considered.
Abstract: We extend to the multidimensional case a Wong-Zakai-type theorem proved by Hu and Oksendal in [7] for scalar quasi-linear Ito stochastic differential equations (SDEs). More precisely, with the aim of approximating the solution of a quasilinear system of Ito's SDEs, we consider for any finite partition of the time interval $[0,T]$ a system of differential equations, where the multidimensional Brownian motion is replaced by its polygonal approximation and the product between diffusion coefficients and smoothed white noise is interpreted as a Wick product. We remark that in the one dimensional case this type of equations can be reduced, by means of a transformation related to the method of characteristics, to the study of a random ordinary differential equation. Here, instead, one is naturally lead to the investigation of a semilinear hyperbolic system of partial differential equations that we utilize for constructing a solution of the Wong-Zakai approximated systems. We show that the law of each element of the approximating sequence solves in the sense of distribution a Fokker-Planck equation and that the sequence converges to the solution of the Ito equation, as the mesh of the partition tends to zero.