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

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  • Damiano Brigo
  • Vladimir Lucic

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.

Suggested Citation

  • Damiano Brigo & Vladimir Lucic, 2026. "Local Stochastic Rough Volatility: Pathwise Filtering and the Conditional Density Equation," Papers 2607.27588, arXiv.org, revised Aug 2026.
    • Handle: RePEc:arx:papers:2607.27588
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    References listed on IDEAS

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    Cited by:

    1. Andrey Itkin, 2026. "A GLE-LSV model with state-dependent leverage and colored noise," Papers 2610.18706, arXiv.org.

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