Functional integration by parts formulae for stochastic Volterra processes

We investigate integration by parts (IBP) formulae for stochastic Volterra equations and we establish the smoothing effect of the expectation. Due to the inherent path-dependent dynamics of this class of processes, standard Bismut--Elworthy--Li (BEL) formulae and lifting procedures fail to produce representations for directional derivatives with respect to the initial curve. We exhibit a new type of fractional IBP for these derivatives which, by means of the Riemann--Liouville fractional derivative, interpolates between the standard chain rule and a pure BEL formula with Cameron--Martin path directions. Our assumptions describe precisely the trade-off between the direction's and the test function's regularities. Crucially, we reveal that more roughness leads to more smoothing: for a power-law kernel with Hurst parameter $H\in(0,1/2)$, we show that the expectation is differentiable along constant directions provided that the test function has H\"older continuity $\beta>2H$. The proof of the formula relies on a careful analysis of the conditional expectation's temporal regularity and on the well-posedness of its Riemann--Liouville derivative. We complement these results with a BEL formula along all square integrable directions whenever the noise is additive, a second order BEL formula and an application to forward and rough volatility models. In the latter case, the derivative is interpreted as the sensitivity with respect to the whole initial forward variance curve.