Weak existence of a solution to a differential equation driven by a very rough fBm

Abstract

We prove that if $f:\mathbb{R}\to \mathbb{R}$ is Lipschitz continuous, then for every $H\in (0,1/4]$ there exists a probability space on which we can construct a fractional Brownian motion $X$ with Hurst parameter $H$, together with a process $Y$ that: (i) is Hölder-continuous with Hölder exponent $\gamma$ for any $\gamma \in (0,H)$; and (ii) solves the differential equation $d{Y}_t=f(Y_t)d{X}_t$. More significantly, we describe the law of the stochastic process $Y$ in terms of the solution to a non-linear stochastic partial differential equation.

20 pages

References in corpus (2)

• Rough Burgers-like equations with multiplicative noise
• A construction of the rough path above fractional Brownian motion using Volterra's representation

Cited by in corpus (2)

• Temporal increments of the KPZ equation with general initial data
• Analysis of the gradient of the solution to a stochastic heat equation via fractional Brownian motion