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Controlled fields, rough stochastic calculus, and Itô-Wentzell-Alekseev-Gröbner identities

Jannis R. Dause, Peter K. Friz, Arnulf Jentzen, Jian Song

Abstract

We develop a calculus of space-time controlled fields for rough stochastic systems. This approach provides a unified composition rule for evaluating random fields along rough semimartingales and yields a rough stochastic Itô-Wentzell formula under natural and verifiable regularity assumptions. Our motivation comes from works of Hudde et al. (2024) and, independently, Del Moral and Singh (2022) where the authors established, respectively, Itô-Alekseev-Gröbner, backward Itô-Wentzell, and diffusion interpolation formulas.

Controlled fields, rough stochastic calculus, and Itô-Wentzell-Alekseev-Gröbner identities

Abstract

We develop a calculus of space-time controlled fields for rough stochastic systems. This approach provides a unified composition rule for evaluating random fields along rough semimartingales and yields a rough stochastic Itô-Wentzell formula under natural and verifiable regularity assumptions. Our motivation comes from works of Hudde et al. (2024) and, independently, Del Moral and Singh (2022) where the authors established, respectively, Itô-Alekseev-Gröbner, backward Itô-Wentzell, and diffusion interpolation formulas.
Paper Structure (26 sections, 22 theorems, 231 equations)

This paper contains 26 sections, 22 theorems, 231 equations.

Table of Contents

  1. Introduction
  2. Space-time rough calculus via controlled fields (\ref{['section:Introduction to controlled fields']})
  3. Moment-free rough stochastic calculus and Itô--Wentzell formula (\ref{['section:On Rough Stochastic Calculus']})
  4. Forward-backward stochastic analysis and stochastic numerics (\ref{['section:Applications to (Rough) Stochastic Analysis']})
  5. Comments and comparisons (\ref{['sec:comments']})
  6. Notation
  7. Linear Algebra:
  8. Probability Theory:
  9. Malliavin Calculus:
  10. Path spaces:
  11. Rough Paths:
  12. Space-time controlled fields
  13. Rough stochastic calculus
  14. Elements of rough semimartingales
  15. Strongly controlled rough semimartingales and paths
  16. ...and 11 more sections

Key Result

Lemma 3.11

Let $\mathfrak{K}\subset W$ closed and $\mathcal{F}=(F, F', \partial F, F", \partial F', \partial^{2} F, \dot{F})$ be a 7-tuple of functions on $[0,T]\times \mathfrak{K}$ of suitable dimension as in def:dim_space_time_controlled. Let us define Then $\mathcal{F}\in \mathscr{D}^{3 \alpha}_{\mathbf{X}} \mathop{\mathrm{\operatorname{Lip}}}\limits^3_{x}(\mathfrak{K}; U)$ iff $[\mathcal{F}]_{\mathfrak{

Theorems & Definitions (70)

  • Definition 3.1
  • Example 3.2
  • Definition 3.3
  • Definition 3.4
  • Remark 3.5
  • Definition 3.6
  • Remark 3.7
  • Remark 3.8
  • Remark 3.9
  • Example 3.10
  • ...and 60 more