Smoothness of Flow and Path-by-Path Uniqueness in Stochastic Differential Equations

Abstract

We consider the stochastic differential equation $$X_t=x_0+{\int }_0^{t}f(X_s)ds+{\int }_0^t\sigma (X_s)d{B}_s^H,$$ with $x_0\in {\mathbb{R}}^d$, $d\ge 1$, $f:{\mathbb{R}}^d\to {\mathbb{R}}^d$ is bounded continuous, $\sigma :{\mathbb{R}}^d\to {\mathbb{R}}^{d\times d}$ is a uniformly elliptic, bounded, twice continuously differentiable conservative vector field and $B^H$ is fractional Brownian motion with $H\in (\frac{1}{3},\frac{1}{2}]$. When $d=1$, $H=\frac{1}{2}$, and $f$ is Hölder continuous, in the spirit of Davie [D07], we establish the existence of a null set $\mathcal{N}$ depending only on $f,\sigma$ such that for all $x_0\in \mathbb{R}$ and $\omega \in \Omega \setminus \mathcal{N}$, the above equation admits a path-by-path unique solution. Our proof is based on establishing the uniform continuous differentiability of the flow associated with the equation. We also establish the path-by-path uniqueness for $d\ge 1$ and $H\in (\frac{1}{3},\frac{1}{2}]$, but the null set may depend on $x_0$, thus extending a result of Catellier-Gubinelli [CG12].

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