  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 H\"older-continuous with H\"older exponent
$\gamma$ for any $\gamma\in(0,H)$; 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.

Probability