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

We prove that if $f:mathbb{R}tomathbb{R}$ is Lipschitz continuous, then for every $Hin(0,1/4]$ 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 Holder-continuous with Holder exponent $gamma$ for any $gammain(0,H)$; and (ii) solves the differential equation $dY_t = f(Y_t) dX_t$. More significantly, we describe the law of the stochastic process $Y$ in terms of the solution to a non-linear stochastic partial differential equation.