Local Stochastic Rough Volatility: Pathwise Filtering and the Conditional Density Equation

Damiano Brigo; Vladimir Lucic

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arXiv · 2607.27588

Local Stochastic Rough Volatility: Pathwise Filtering and the Conditional Density Equation

Damiano Brigo·Vladimir Lucic

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Abstract

This article studies the conditional-density equation and its pathwise transformation in local stochastic rough volatility models, with rough Heston (rHeston) as the main explicit example. Under the stated common-filtration, measurability, predictability and spatial-regularity assumptions, we show that the Ito-Wentzell random-PDE reduction of the conditional density SPDE remains valid under local stochastic rough volatility. After fixing a common-environment realization and the associated stochastic flow, the transformed equation becomes a deterministic PDE with path-dependent coefficients. This yields a pathwise Fokker--Planck formulation that connects naturally with Rao--Blackwellized calibration. In the pure rough Heston case, the transformed coefficients simplify and the conditional density admits an explicit lognormal form.

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Damiano Brigo, Vladimir Lucic. 2026-07-30. Local Stochastic Rough Volatility: Pathwise Filtering and the Conditional Density Equation. https://arxiv.org/abs/2607.27588

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