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In this paper we study dynamic backward problems, with the computation ofconditional expectations as a main objective, in a framework where the(forward) state process satisfies a Volterra type SDE, with fractional Brownianmotion as a typical example. Such processes are neither Markov processes norsemimartingales, and most notably, they feature a certain time inconsistencywhich makes any direct application of Markovian ideas, such as flow properties,impossible without passing to a path-dependent framework. Our main result is afunctional It\^{o} formula, extending the seminal work of Dupire \cite{Dupire}to our more general framework. In particular, unlike in \cite{Dupire} where oneneeds only to consider the stopped paths, here we need to concatenate theobserved path up to the current time with a certain smooth observable curvederived from the distribution of the future paths. This new feature is due tothe time inconsistency involved in this paper. We then derive the pathdependent PDEs for the backward problems. Finally, an application to optionpricing in a financial market with rough volatility is presented.

A Martingale Approach for Fractional Brownian Motions and Related Path Dependent PDEs