  In this paper we study dynamic backward problems, with the computation of
conditional expectations as a main objective, in a framework where the
(forward) state process satisfies a Volterra type SDE, with fractional Brownian
motion as a typical example. Such processes are neither Markov processes nor
semimartingales, and most notably, they feature a certain time inconsistency
which makes any direct application of Markovian ideas, such as flow properties,
impossible without passing to a path-dependent framework. Our main result is a
functional It\^{o} formula, extending the seminal work of Dupire \cite{Dupire}
to our more general framework. In particular, unlike in \cite{Dupire} where one
needs only to consider the stopped paths, here we need to concatenate the
observed path up to the current time with a certain smooth observable curve
derived from the distribution of the future paths. This new feature is due to
the time inconsistency involved in this paper. We then derive the path
dependent PDEs for the backward problems. Finally, an application to option
pricing in a financial market with rough volatility is presented.
