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Markovian Lifts of Stochastic Volterra Equations in Sobolev Spaces: Solution theory, an Ito Formula and Invariant Measures

Florian Huber

TL;DR

This work develops a Markovian lift for stochastic Volterra equations with completely monotone kernels by embedding the memoryful SVE into a stochastic evolution equation on weighted Sobolev spaces. The lift enables a rigorous solution theory, with existence and uniqueness established first under Lipschitz conditions and then extended to general coefficients via approximation and tightness arguments, complemented by strong a-priori estimates and time-regularity. It also proves an Itô-type formula for the lifted system and shows the existence of invariant measures for both the lifted SEE and the original SVE under suitable assumptions, leveraging weak (generalized) Feller properties. The framework provides a unified approach to analyze long-term behavior of SVEs with memory, with potential applications to rough volatility and other memory-driven systems, and lays a foundation for further duality and renewal-type analyses via the lifted dynamics in weighted Sobolev spaces.

Abstract

We investigate Markovian lifts of stochastic Volterra equations (SVEs) with completely monotone kernels and general coefficients within a class of weighted Sobolev spaces. Our primary focus is developing a comprehensive solution theory for a class of non-local stochastic evolution equations (SEEs) encompassing these Markovian lifts. This enables us to provide conditions for the existence of invariant measures for the lifted processes and the corresponding SVE. Another key contribution is an Ito-type formula for the stochastic Volterra equations under consideration.

Markovian Lifts of Stochastic Volterra Equations in Sobolev Spaces: Solution theory, an Ito Formula and Invariant Measures

TL;DR

This work develops a Markovian lift for stochastic Volterra equations with completely monotone kernels by embedding the memoryful SVE into a stochastic evolution equation on weighted Sobolev spaces. The lift enables a rigorous solution theory, with existence and uniqueness established first under Lipschitz conditions and then extended to general coefficients via approximation and tightness arguments, complemented by strong a-priori estimates and time-regularity. It also proves an Itô-type formula for the lifted system and shows the existence of invariant measures for both the lifted SEE and the original SVE under suitable assumptions, leveraging weak (generalized) Feller properties. The framework provides a unified approach to analyze long-term behavior of SVEs with memory, with potential applications to rough volatility and other memory-driven systems, and lays a foundation for further duality and renewal-type analyses via the lifted dynamics in weighted Sobolev spaces.

Abstract

We investigate Markovian lifts of stochastic Volterra equations (SVEs) with completely monotone kernels and general coefficients within a class of weighted Sobolev spaces. Our primary focus is developing a comprehensive solution theory for a class of non-local stochastic evolution equations (SEEs) encompassing these Markovian lifts. This enables us to provide conditions for the existence of invariant measures for the lifted processes and the corresponding SVE. Another key contribution is an Ito-type formula for the stochastic Volterra equations under consideration.
Paper Structure (16 sections, 42 theorems, 143 equations)

This paper contains 16 sections, 42 theorems, 143 equations.

Table of Contents

  1. introduction and main results
  2. Structure of the paper
  3. Notation
  4. Preliminaries: Weighted spaces
  5. The lifted equation
  6. The operator and semigroup
  7. Estimates on Hilbert spaces
  8. Existence
  9. Step 1: The Lipschitz case
  10. Step 2: General coefficients
  11. A-priori estimates
  12. Long term existence and invariant measures
  13. Ito-formula for Volterra equations
  14. Appendix
  15. Proofs of technical results
  16. ...and 1 more sections

Key Result

Theorem 2

schilling2009bernstein. The following three assertions are equivalent:

Theorems & Definitions (111)

  • Definition 1
  • Theorem 2
  • Definition 3
  • Definition 4
  • Theorem 5
  • Theorem 6
  • Remark 7
  • Theorem 8
  • Theorem 9
  • Proposition 10
  • ...and 101 more