Paper: arXiv 2509.21608

Authors: Ioannis Gasteratos, Alexandre Pannier

Abstract

We associate backward and forward Kolmogorov equations to a class of fully nonlinear Stochastic Volterra Equations (SVEs) with convolution kernels $K$ that are singular at the origin. Working on a carefully chosen Hilbert space $\mathcal{H}_1$, we rigorously establish a link between solutions of SVEs and Markovian mild solutions of a Stochastic Partial Differential Equation (SPDE) of transport-type. Then, we obtain two novel Itô formulae for functionals of mild solutions and, as a byproduct, show that their laws solve corresponding Fokker-Planck equations. Finally, we introduce a natural notion of “singular” directional derivatives along $K$ and prove that (conditional) expectations of SVE solutions can be expressed in terms of the unique solution to a backward Kolmogorov equation on $\mathcal{H}_1$. Our analysis relies on stochastic calculus in Hilbert spaces, the reproducing kernel property of the state space $\mathcal{H}_1,$ as well as crucial invariance and smoothing properties that are specific to the SPDEs of interest. In the special case of singular power-law kernels, our conditions guarantee well-posedness of the backward equation either for all values of the Hurst parameter $H,$ when the noise is additive, or for all $H>1/4$ when the noise is multiplicative.

Complexity vs Empirical Score

  • Math Complexity: 9.5/10
  • Empirical Rigor: 1.5/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper provides a highly sophisticated, rigorous mathematical framework using infinite-dimensional stochastic analysis and SPDEs to solve a long-standing open problem regarding Kolmogorov equations for singular SVEs (relevant to rough volatility). It is purely theoretical with no empirical backtests or market data analysis, but its mathematical contribution to quantitative finance theory is exceptional.

Research Flowchart

  flowchart TD
    A[Goal: Formulate Backward & Forward Kolmogorov Equations for Singular SVEs] --> B[Input: Singular Kernel SVEs & State Space H1]
    B --> C[Lift SVEs to Transport-Type SPDEs in H1]
    C --> D[Develop Itô Formulae & Infinite-Dimensional Stochastic Calculus]
    D --> E[Derive Fokker-Planck Equations for SVE Laws]
    D --> F[Solve Backward Kolmogorov Equation using Singular Directional Derivatives]
    E --> G[Outcomes: Rigorous PDE/SPDE Link & Well-Posedness for H > 1/4]
    F --> G

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