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We investigate the well-posedness of stochastic differential equations driven by fractional Brownian motion, focusing on the long-range dependent case $H \in (\frac{1}{2}, 1)$. While existing results on regularization by such noise…
We study a two-dimensional stochastic differential equation that has a unique weak solution but no strong solution. We show that this SDE shares notable properties with Tsirelson's example of a one-dimensional SDE with no strong solution.…
We prove an It\^o-Wentzell formula for the fractional Brownian motion. As an application we derive an existence and uniqueness result for a class of stochastic differential equations driven by this stochastic process.
In this paper we shall establish an existence and uniqueness result for solutions of multidimensional, time dependent, stochastic differential equations driven simultaneously by a multidimensional fractional Brownian motion with Hurst…
We propose a new numerical method for one dimensional stochastic differential equations (SDEs). The main idea of this method is based on a representation of a weak solution of a SDE with a time changed Brownian motion, dated back to Doeblin…
In this paper, we investigate the stochastic differential equation on $\mathbb{R}^d,d\geq2$: \begin{align*} \dif X_t&=v(t,X_t)\dif t+\sqrt{2} \dif W_t. \end{align*} For any finite collection of initial probability measures…
In this paper we investigate classical solution of a semi-linear system of backward stochastic integral partial differential equations driven by a Brownian motion and a Poisson point process. By proving an It\^{o}-Wentzell formula for jump…
We present an algorithm for solving stochastic heat equations, whose key ingredient is a non-uniform time discretization of the driving Brownian motion $W$. For this algorithm we derive an error bound in terms of its number of evaluations…
In this work, we will show the existence and uniqueness of the solution to the semi linear stochastic differential equations driven by weighted fractional Brownian motion with delay. We also prove smoothness of the density of the solution…
We consider the equations of motion for an incompressible Non-Newtonian fluid in a bounded Lipschitz domain $G\subset\mathbb R^d$ during the time intervall $(0,T)$ together with a stochastic perturbation driven by a Brownian motion $W$. The…
In this note we prove an existence and uniqueness result for the solution of multidimensional stochastic delay differential equations with normal reflection. The equations are driven by a fractional Brownian motion with Hurst parameter…
We are interested in existence of solutions to the $d$-dimensional equation \begin{equation*} X_t=x_0+\int_0^t b(X_s)ds + B_t, \end{equation*} where $B$ is a (fractional) Brownian motion with Hurst parameter $H\leqslant 1/2$ and $b$ is an…
Let $W$ denote the Brownian motion. For any exponentially bounded Borel function $g$ the function $u$ defined by $u(t,x)= \mathbb{E}[g(x{+}\sigma W_{T-t})]$ is the stochastic solution of the backward heat equation with terminal condition…
We study the long-time behaviour of solutions to a class of $d$-dimensional stochastic differential equations driven by fractional Brownian motion with Hurst parameter $H \in (0,1)$. The drift consists of a dissipative Lipschitz term and a…
In this paper, we are interested in the following singular stochastic differential equation (SDE) $${\rm d} X_t = b(t,X_t) {\rm d} t + {\rm d} B_{t},\ 0\leq t\leq T,\ X_0 = x \in \mathbb{R}^d,$$ where the drift coefficient $b:[0,T]\times…
In this paper, we study a class of multi-dimensional reflected backward stochastic differential equations when the noise is driven by a Brownian motion and an independent Poisson point process, and when the solution is forced to stay in a…
We prove the existence and uniqueness of mild solution for the stochastic partial differential equation $$\left(\partial^\alpha - \textit{B} \right) u(t,x)= u(t,x) \cdot \dot{W}(t,x),$$ where $$\alpha \in (1/2, 1)\cup(1, 2);$$ $\textit{B}$…
We show that the only flow solving the stochastic differential equation (SDE) on $\RR$ $$dX_t = 1_{\{X_t>0\}}W_+(dt) + 1_{\{X_t<0\}}dW_-(dt),$$ where $W^+$ and $W^-$ are two independent white noises, is a coalescing flow we will denote…
We study efficiency of non-parametric estimation of diffusions (stochastic differential equations driven by Brownian motion) from long stationary trajectories. First, we introduce estimators based on conditional expectation which is…
In this paper, we consider a stochastic differential equation driven by a fractional Brownian motion (fBm) and a Wiener process and having jumps. We prove that this equation has a unique solution and show that all its moments are finite.