Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Quadratic Wiener functionals -- transformations and quadratic forms

Setsuo Taniguchi

Abstract

Quadratic Wiener functionals are investigated systematically through transformations of order one on the Wiener space with the help of Malliavin calculus. The bi-directional relationship between quadratic Wiener functionals and transformations of order one is established via change of variables formulas on the Wiener space. The relationship is applied to the investigation of Laplace transformations of quadratic Wiener functionals. This note is made due to establishing a systematic framework to study quadratic Wiener functionals and revisiting the past works by the author with the framework.

Quadratic Wiener functionals -- transformations and quadratic forms

Abstract

Quadratic Wiener functionals are investigated systematically through transformations of order one on the Wiener space with the help of Malliavin calculus. The bi-directional relationship between quadratic Wiener functionals and transformations of order one is established via change of variables formulas on the Wiener space. The relationship is applied to the investigation of Laplace transformations of quadratic Wiener functionals. This note is made due to establishing a systematic framework to study quadratic Wiener functionals and revisiting the past works by the author with the framework.
Paper Structure (30 sections, 77 theorems, 1164 equations)

This paper contains 30 sections, 77 theorems, 1164 equations.

Table of Contents

  1. Quadratic forms
  2. Series expansion of quadratic forms
  3. Series expansions due to Lévy and Kac
  4. From transformations to quadratic forms
  5. Transformations of order one
  6. Inverse transformation of order one
  7. Linear transformations
  8. From quadratic forms to transformations
  9. Surjectivity --- finding $\kappa$ with $\eta(\kappa)=\eta$
  10. Bijectivity
  11. Quadratic Wiener functional of harmonic-oscillator type
  12. Derivation of the Girsanov theorem
  13. Linear adapted transformations
  14. Linear adapted transformation
  15. Feynman-Kac density function
  16. ...and 15 more sections

Key Result

Theorem (1.1.1)

Let $\eta\in\mathscr{S}_2$. (i) It holds that where the right term is defined by regarding $B_\eta\in \mathcal{S}({\mathcal{H}^{\otimes 2}})$ as a constant Wiener functional in $\mathbb{D}^\infty({\mathcal{H}^{\otimes 2}})$ and applying ${\mathrm{D}}^*$ twice. (ii) For any orthonormal basis (ONB in short) $\{h_n\}_{n=1}^\infty$ of $\mathcal{H}$ where the series converges in $L^p(\mu)$ for every $

Theorems & Definitions (198)

  • Theorem (1.1.1)
  • Lemma (1.1.2)
  • proof
  • proof : Proof of Theorem \ref{['t.q.eta']}
  • Theorem (1.1.3)
  • Lemma (1.1.4)
  • proof
  • proof : Proof of Theorem \ref{['t.w.chaos']}
  • Proposition (1.1.5)
  • proof
  • ...and 188 more