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Stochastic Volterra equations with random functional coefficients in Banach spaces

Alexander Kalinin

TL;DR

The paper develops a Banach-space theory for stochastic Volterra equations with random, including distribution-dependent, coefficients and singular kernels. It introduces a weak modification framework and leverages Wasserstein-measurability to prove existence and uniqueness of strong solutions under moment-growth and Lipschitz-type conditions. Two solution notions are developed: locally bounded $p$-th moment solutions and locally $p$-fold integrable moment solutions, with strong results in controlled and McKean–Vlasov settings. The approach combines new moment-inequality tools for iterated kernels, measurability of distribution maps, and Picard-iteration schemes to extend stochastic Volterra theory to Banach spaces with random coefficients and kernel singularities, with implications for rough volatility modelling and stochastic optimization.

Abstract

We derive unique Banach-valued solutions to stochastic Volterra equations with random coefficients that may depend on pure chance and involve singular kernels. In particular, for controlled and distribution-dependent coefficients these solutions become strong, as a measurability analysis of the Wasserstein metric confirms. The presented novel approach is based on the proof that a stochastic Volterra integral admits a progressively measurable modification in a weak sense and on sharp moment estimates for non-negative product measurable processes.

Stochastic Volterra equations with random functional coefficients in Banach spaces

TL;DR

The paper develops a Banach-space theory for stochastic Volterra equations with random, including distribution-dependent, coefficients and singular kernels. It introduces a weak modification framework and leverages Wasserstein-measurability to prove existence and uniqueness of strong solutions under moment-growth and Lipschitz-type conditions. Two solution notions are developed: locally bounded -th moment solutions and locally -fold integrable moment solutions, with strong results in controlled and McKean–Vlasov settings. The approach combines new moment-inequality tools for iterated kernels, measurability of distribution maps, and Picard-iteration schemes to extend stochastic Volterra theory to Banach spaces with random coefficients and kernel singularities, with implications for rough volatility modelling and stochastic optimization.

Abstract

We derive unique Banach-valued solutions to stochastic Volterra equations with random coefficients that may depend on pure chance and involve singular kernels. In particular, for controlled and distribution-dependent coefficients these solutions become strong, as a measurability analysis of the Wasserstein metric confirms. The presented novel approach is based on the proof that a stochastic Volterra integral admits a progressively measurable modification in a weak sense and on sharp moment estimates for non-negative product measurable processes.
Paper Structure (20 sections, 34 theorems, 162 equations)

This paper contains 20 sections, 34 theorems, 162 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and main results
  3. Iterated kernels and resolvents
  4. Stochastic Volterra integrals in Banach spaces
  5. Admissible coefficients and notions of solutions
  6. Solutions with locally essentially bounded moment functions
  7. Solutions with locally integrable moment functions
  8. General results on stochastic processes
  9. Weakly modified processes
  10. Approximations of processes and distribution maps
  11. Sharp moment and integral inequalities
  12. Sharp moment estimates for stochastic Volterra processes
  13. Supplementary analytic results
  14. Integral estimates for iterated kernels
  15. A metrical decomposition principle
  16. ...and 5 more sections

Key Result

Proposition 2.4

Let $U:I\times I\times\Omega\rightarrow E$ be $\mathcal{B}(I)\otimes\mathcal{A}$-measurable and $\mu$ be a Borel measure on $I$ such that $\mu([0,t])$ and $\int_{[0,t]}|U_{t,s}|\,\mu(\mathrm{d}s)$ are finite for all $t\in I$. Then is an $\mathbb{F}$-progressively measurable process.

Theorems & Definitions (101)

  • Definition 2.1
  • Remark 2.2
  • Example 2.3
  • Proposition 2.4
  • Example 2.5: Hilbert-Schmidt norm
  • Example 2.6
  • Proposition 2.7
  • Remark 2.8
  • Proposition 2.9
  • Definition 2.10
  • ...and 91 more