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The $C^{0,1}$ Itô-Ventzell formula for weak Dirichlet processes

Felix Fießinger, Mitja Stadje

TL;DR

The paper extends the Itô-Ventzell formula to continuous weak Dirichlet processes driven by stochastic flows in $C^{0,1}$, reducing spatial regularity requirements from $C^{2}$ to $C^{0,1}$ under mild integrability assumptions. It proves a decomposed representation for $F(t,X_t)$ that includes a linear, continuous weak-zero-energy remainder $\mathcal{B}^X(F)$ and preserves the weak Dirichlet structure, connecting stochastic flows to SPDEs and financial models. The authors derive a novel quadratic-variation formula, provide a representation result for strong solutions of time-dependent stochastic elliptic PDEs, and illustrate the framework with a large-investor price-impact example. These results broaden stochastic calculus with rough flows and offer new tools for SPDE analysis and finance under weaker regularity constraints.

Abstract

This paper proves an extension of the Itô-Ventzell formula that applies to stochastic flows in $C^{0,1}$ for continuous weak Dirichlet processes. We apply this theorem, for example, to give a representation result for strong solutions of time-dependent elliptic SPDEs, to derive formulas for quadratic variations, and to relax assumptions in a financial mathematics context.

The $C^{0,1}$ Itô-Ventzell formula for weak Dirichlet processes

TL;DR

The paper extends the Itô-Ventzell formula to continuous weak Dirichlet processes driven by stochastic flows in , reducing spatial regularity requirements from to under mild integrability assumptions. It proves a decomposed representation for that includes a linear, continuous weak-zero-energy remainder and preserves the weak Dirichlet structure, connecting stochastic flows to SPDEs and financial models. The authors derive a novel quadratic-variation formula, provide a representation result for strong solutions of time-dependent stochastic elliptic PDEs, and illustrate the framework with a large-investor price-impact example. These results broaden stochastic calculus with rough flows and offer new tools for SPDE analysis and finance under weaker regularity constraints.

Abstract

This paper proves an extension of the Itô-Ventzell formula that applies to stochastic flows in for continuous weak Dirichlet processes. We apply this theorem, for example, to give a representation result for strong solutions of time-dependent elliptic SPDEs, to derive formulas for quadratic variations, and to relax assumptions in a financial mathematics context.
Paper Structure (8 sections, 12 theorems, 32 equations)

This paper contains 8 sections, 12 theorems, 32 equations.

Table of Contents

  1. Introduction
  2. Basic Definitions
  3. Main Result
  4. Proof of Theorem \ref{['th: main result']}
  5. Applications
  6. Formulas for quadratic variations
  7. A representation result for strong solutions of SPDEs
  8. An example from financial mathematics

Key Result

Theorem 3.1

Let $X$ be a continuous weak Dirichlet FQV process and $F \in C^{0,1}$ be a stochastic flow with local characteristics $(\beta(t,x),\gamma(t,x))$. Furthermore, we assume: Then it holds for all $t \in [0,T]$: where $\mathcal{B}^X$ is linear and continuous (in $F$) with respect to the topology of $ucp$ convergence and $F(t,X_t)$ is again a continuous weak Dirichlet process. Moreover,

Theorems & Definitions (34)

  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Definition 2.4
  • Definition 2.5
  • Example 2.6
  • Definition 2.7
  • Remark 2.8
  • Theorem 3.1
  • Corollary 3.2
  • ...and 24 more