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An introduction to Malliavin calculus

Luciano Tubaro, Margherita Zanella

TL;DR

The notes provide a structured, Gaussian-centric introduction to Malliavin calculus, highlighting how core operators $D$, $\delta$, and $L$ interplay within the OU framework and Wiener chaos. They develop the general theory and then contrast two prominent topological-space derivatives, Bogachev's $\nabla_H$ and Da Prato's $Q^{1/2}\nabla$, showing they yield the same domain yet arise from different choices of the underlying Hilbert structures. The Wiener chaos decomposition and Hermite polynomials serve as foundational tools, enabling a precise analysis of densities via duality and Malliavin matrices. The treatment emphasizes the Cameron–Martin space and Gaussian measures, illustrating how differentiability concepts extend to infinite-dimensional Banach and Hilbert spaces and informing applications to density regularity and stochastic analysis.

Abstract

These Lecture Notes are a brief introduction to the Malliavin calculus. In particular, different notions of Malliavin derivative found in the literature are considered and compared.

An introduction to Malliavin calculus

TL;DR

The notes provide a structured, Gaussian-centric introduction to Malliavin calculus, highlighting how core operators , , and interplay within the OU framework and Wiener chaos. They develop the general theory and then contrast two prominent topological-space derivatives, Bogachev's and Da Prato's , showing they yield the same domain yet arise from different choices of the underlying Hilbert structures. The Wiener chaos decomposition and Hermite polynomials serve as foundational tools, enabling a precise analysis of densities via duality and Malliavin matrices. The treatment emphasizes the Cameron–Martin space and Gaussian measures, illustrating how differentiability concepts extend to infinite-dimensional Banach and Hilbert spaces and informing applications to density regularity and stochastic analysis.

Abstract

These Lecture Notes are a brief introduction to the Malliavin calculus. In particular, different notions of Malliavin derivative found in the literature are considered and compared.

Paper Structure

This paper contains 46 sections, 72 theorems, 295 equations.

Table of Contents

  1. Introduction
  2. Gaussian Hilbert spaces
  3. Preliminaries
  4. Gaussian random variables
  5. Gaussian Hilbert spaces
  6. The Isserlis formula
  7. The Wiener Chaos Decomposition
  8. Hermite Polynomials
  9. The Ornstein-Uhlenbeck semigroup and its infinitesimal generator
  10. The semigroup of Ornstein-Uhlenbeck
  11. The generator of the Ornstein-Uhlenbeck semigroup
  12. The Malliavin derivative
  13. The Malliavin derivative of real-valued random variables
  14. The Malliavin derivative of Hilbert-valued random variables
  15. The directional Malliavin derivative
  16. ...and 31 more sections

Key Result

Proposition 2.5

Any set of random variables in a Gaussian linear space has a joint normal distribution.

Theorems & Definitions (197)

  • Definition 2.1: Gaussian measures on $\mathbb{R}$
  • Definition 2.2: Real-valued Gaussian random variable
  • Definition 2.3
  • Definition 2.4
  • Proposition 2.5
  • proof
  • Theorem 2.6
  • proof
  • Proposition 2.7
  • proof
  • ...and 187 more