Functional integration by parts formulae for stochastic Volterra processes

2026-05-28

AI Summary

This work addresses the failure of the classical Bismut–Elworthy–Li (BEL) formula for stochastic Volterra processes due to their path-dependent nature. To overcome this, the authors develop a novel integration-by-parts (IBP) formula based on Riemann–Liouville fractional derivatives, which interpolates between the standard chain rule and the BEL formula. Their analysis reveals a counterintuitive smoothing effect: the rougher the noise—characterized by a Hurst parameter $H \in (0,1/2)$—the smoother the resulting expectation functional. Specifically, directional differentiability along constant directions is guaranteed whenever the test function’s Hölder exponent satisfies $\beta > 2H$. The framework is further extended to establish first- and second-order BEL formulas for square-integrable directions in additive noise settings, with applications to forward and rough volatility models that clarify the trade-off between the regularity of the test function and the existence of directional derivatives.

Abstract

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 Hölder continuity $β>2H$. 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.

Problem

• stochastic Volterra processes
• integration by parts
• path-dependent dynamics
• directional derivatives
• initial curve

Innovation

• fractional integration by parts
• stochastic Volterra processes
• Bismut–Elworthy–Li formula
• Riemann–Liouville derivative
• rough volatility