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Regular occupation measures of Volterra processes

Martin Friesen

Abstract

We give a local non-determinism condition applicable to general Volterra Ito processes that allow us to obtain the space-time regularity of the occupation and self-intersection measures. For the particular case of solutions to a stochastic Volterra equation, we also obtain the one-dimensional distributions' regularity and study the finite-dimensional distributions' absolute continuity. Finally, based on the previously shown regularity of the self-intersection measure, we prove the existence, uniqueness and stability of stochastic equations with distributional drifts of "self-intersection" type in terms of corresponding two-parameter nonlinear Young equations.

Regular occupation measures of Volterra processes

Abstract

We give a local non-determinism condition applicable to general Volterra Ito processes that allow us to obtain the space-time regularity of the occupation and self-intersection measures. For the particular case of solutions to a stochastic Volterra equation, we also obtain the one-dimensional distributions' regularity and study the finite-dimensional distributions' absolute continuity. Finally, based on the previously shown regularity of the self-intersection measure, we prove the existence, uniqueness and stability of stochastic equations with distributional drifts of "self-intersection" type in terms of corresponding two-parameter nonlinear Young equations.
Paper Structure (15 sections, 14 theorems, 174 equations)

This paper contains 15 sections, 14 theorems, 174 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Function spaces
  4. Stochastic sewing techniques
  5. Volterra Ito-processes
  6. Regularity of weighted occupation measures
  7. Regularity of the law
  8. Examples
  9. Stochastic Volterra equations
  10. Regularity
  11. Finite dimensional distributions
  12. Regularization by noise for rough drifts with self-intersections
  13. Nonlinear Young integration
  14. Rough drift of self-intersection type
  15. Subordination principle

Key Result

Lemma 2.1

If $s \in \mathbb{R}$ and $1 < p \leq 2$, then while for $p = 1$ we obtain $\mathcal{F}L_1^s(\mathbb{R}^d) \hookrightarrow \mathcal{C}^s(\mathbb{R}^d)$.

Theorems & Definitions (45)

  • Lemma 2.1
  • proof
  • Lemma 2.2
  • Proposition 2.3
  • proof
  • Remark 3.1
  • Example 3.2
  • Theorem 3.3
  • proof
  • Remark 3.4
  • ...and 35 more