Local Stochastic Rough Volatility: Pathwise Filtering and the Conditional Density Equation
arXiv:2607.27588
The paper analyzes the conditional density equation for local stochastic rough volatility models, showing that the Itô‑Wentzell random PDE reduction remains valid and can be transformed into a deterministic pathwise Fokker‑Planck PDE, with explicit lognormal solutions for the rough Heston case.
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.