Related papers: Rough differential equations in the flow approach
This paper studies a stochastic functional differential equation driven by a fractional Brownian motion with Hurst parameter H>1/2, constrained to be reflected at 0. We prove the existence of solutions using the Euler method. However,…
In this article we study a class of singular stochastic differential equations driven by fractional Brownian motion with Hurst parameter H<1/2. The solution is constructed as the limit of a family of approximating processes, and its…
A theory of differential equations driven by a non-differentiable path has recently been developed by Lyons. We develop an alternative approach to this theory, using (modified Euler approximations), and investigate its applicability to…
Strongly consistent and asymptotic normal estimators of the Hurst index of a stochastic differential equation driven by a fractional Brownian motion are proposed. The estimators are based on discrete observations of the underlying process.
We consider nonlinear parabolic evolution equations of the form $\partial_{t}u=F(t,x,Du,D^{2}u) $, subject to noise of the form $H(x,Du) \circ dB$ where $H$ is linear in $Du$ and $\circ dB$ denotes the Stratonovich differential of a…
We study the approximation of stochastic differential equations driven by a fractional Brownian motion with Hurst parameter $H>1/2$. For the mean-square error at a single point we derive the optimal rate of convergence that can be achieved…
This note is devoted to show how to push forward the algebraic integration setting in order to treat differential systems driven by a noisy input with H\"older regularity greater than 1/4. After recalling how to treat the case of ordinary…
This paper is devoted to the synchronization of stochastic differential equations driven by the linear multiplicative fractional Brownian motion with Hurst parameter $H\in(\frac{1}{2},1)$. We firstly prove that the equation has a unique…
In this paper we study a singular stochastic differential equation driven by an additive fractional Brownian motion with Hurst parameter $H>\frac 12$. Under some assumptions on the drift, we show that there is a unique solution, which has…
We consider the problem of Hurst index estimation for solutions of stochastic differential equations driven by an additive fractional Brownian motion. Using techniques of the Malliavin calculus, we analyze the asymptotic behavior of the…
These are lecture notes for a Master 2 course on rough differential equations driven by weak geometric Holder p-rough paths, for any p>2. They provide a short, self-contained and pedagogical account of the theory, with an emphasis on flows.…
In this note we prove the existence of a density for the law of the solution for 1-dimensional stochastic delay differential equations with normal reflection. The equations are driven by a fractional Brownian motion with Hurst parameter $H…
Motivated by the probabilistic representation for solutions of the Navier-Stokes equations, we introduce a novel class of stochastic differential equations that depend on the entire flow of its time marginals. We establish the existence and…
This paper is devoted to studying the averaging principle for fast-slow system of rough differential equations driven by mixed fractional Brownian rough path. The fast component is driven by Brownian motion, while the slow component is…
Given a fractional Brownian motion \,\,$(B_{t}^{H})_{t\geq 0}$,\, with Hurst parameter \,$> 1/2$\,\,we study the properties of all solutions of \,\,: {equation} X_{t}=B_{t}^{H}+\int_0^t X_{u}d\mu(u), \;\; 0\leq t\leq 1{equation} A different…
Fourier normal ordering \cite{Unt09bis} is a new algorithm to construct explicit rough paths over arbitrary H\"older-continuous multidimensional paths. We apply in this article the Fourier normal ordering ordering algorithm to the…
We prove that a stochastic flow of reflected Brownian motions in a smooth multidimensional domain is differentiable with respect to its initial position. The derivative is a linear map represented by a multiplicative functional for…
In the paper, Harnack inequalities are established for stochastic differential equations driven by fractional Brownian motion with Hurst parameter $H<1/2$. As applications, strong Feller property, log-Harnack inequality and entropy-cost…
In this paper, we derive the exact rate of convergence of some approximation schemes associated to scalar stochastic differential equations driven by a fractional Brownian motion with Hurst index H.
In this paper, we study a class of dissipative stochastic differential equations driven by nonlinear multiplicative fractional Brownian noise with Hurst index $H \in \left(\frac{1}{3},\frac{1}{2})\cup(\frac{1}{2}, 1\right) $. We establish…