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Malliavin Calculus for rough stochastic differential equations

Fabio Bugini, Michele Coghi, Torstein Nilssen

Abstract

In this work we show that rough stochastic differential equations (RSDEs), as introduced by Friz, Hocquet, and Lê (2021), are Malliavin differentiable. We use this to prove existence of a density when the diffusion coefficients satisfies standard ellipticity assumptions. Moreover, when the coefficients are smooth and the diffusion coefficients satisfies a Hörmander condition, the density is shown to be smooth. The key ingredient is to develop a comprehensive theory of linear rough stochastic differential equations, which could be of independent interest.

Malliavin Calculus for rough stochastic differential equations

Abstract

In this work we show that rough stochastic differential equations (RSDEs), as introduced by Friz, Hocquet, and Lê (2021), are Malliavin differentiable. We use this to prove existence of a density when the diffusion coefficients satisfies standard ellipticity assumptions. Moreover, when the coefficients are smooth and the diffusion coefficients satisfies a Hörmander condition, the density is shown to be smooth. The key ingredient is to develop a comprehensive theory of linear rough stochastic differential equations, which could be of independent interest.
Paper Structure (14 sections, 36 theorems, 306 equations)

This paper contains 14 sections, 36 theorems, 306 equations.

Table of Contents

  1. Introduction
  2. Notation and Preliminaries
  3. Notation
  4. Rough paths theory
  5. Preliminaries on Malliavin Calculus
  6. Rough stochastic analysis
  7. Rough Itô formula for functions of polynomial growth
  8. Linear rough stochastic differential equations
  9. Malliavin differentiability of the solutions
  10. Hörmander-type results
  11. Existence of densities under Hörmander condition
  12. Smoothness of densities under Hörmander condition
  13. Weighted norms
  14. Malliavin calculus for smooth RSDEs

Key Result

Theorem 1

If $b$ and $\sigma$ are differentiable and bounded and $\beta$ is three times differentiable and bounded, then the solution $X$ to equation eq:RSDE_intro is Malliavin differentiable.

Theorems & Definitions (88)

  • Theorem : see Theorem \ref{['thm:malliavincalculusforRSDEs']}
  • Theorem : see Theorem \ref{['thm:wellposednesslinearRSDEs']}
  • Theorem
  • Remark 1.2
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Theorem 2.6: Stochastic sewing lemma
  • ...and 78 more