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

2026-07-29

AI Summary

This study investigates the evolution of conditional densities and pathwise filtering under a locally stochastic rough volatility model. By conditioning on a fixed environmental path, the underlying stochastic partial differential equation is transformed into a deterministic PDE with path-dependent coefficients. The authors extend the Itô–Wentzell formula to this setting for the first time, thereby establishing a pathwise Fokker–Planck framework. In the pure rough Heston case, they derive an explicit log-normal analytical solution for the conditional density. This work not only uncovers an intrinsic connection between path-conditioned densities and Rao–Blackwellized calibration but also provides a theoretical foundation and computational tools for efficient parameter calibration and filtering.

Abstract

This note 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 Itô-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.

Problem

• rough volatility
• conditional density
• stochastic PDE
• pathwise filtering
• rough Heston

Innovation

• rough volatility
• conditional density SPDE
• Itô-Wentzell formula
• pathwise Fokker–Planck equation
• Rao–Blackwellized calibration