  We provide explicit series expansions to certain stochastic path-dependent
integral equations in terms of the path signature of the time augmented driving
Brownian motion. Our framework encompasses a large class of stochastic linear
Volterra and delay equations and in particular the fractional Brownian motion
with a Hurst index $H \in (0, 1)$. Our expressions allow to disentangle an
infinite dimensional Markovian structure and open the door to straightforward
and simple approximation schemes, that we illustrate numerically.
