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Stochastic Calculus for Rough Fractional Brownian Motion via Operator Factorization

Ramiro Fontes

TL;DR

This work develops an operator-theoretic stochastic calculus for rough fractional Brownian motion with $H<\frac{1}{2}$ by building an operator-covariant derivative $D_X$ as the adjoint of a divergence and introducing a predictable projection $\Pi_X$ in the Cameron–Martin (RKHS) space. A central result is the factorization $(\mathrm{Id}-\mathbb{E})F=\delta_X(\Pi_X D_X F)$, which yields explicit first-order derivative formulas for functionals and a controlled martingale expansion that mirrors rough-path structure. The predictable component $\Pi_X D_X F$ is identified with the Gubinelli derivative, linking the operator framework to rough-path theory while avoiding the construction of iterated integrals. The approach extends naturally to mixed semimartingale–rough processes, enabling unified stochastic calculus for models combining Brownian and rough components, with concrete applications to optimal filtering, rough-volatility hedging, and SDEs driven by rough noise, along with practical numerical schemes based on RKHS truncation. This framework provides conceptual clarity and computational advantages by grounding rough-stochastic calculus in Gaussian energy geometry and Malliavin-type adjointness, offering a first-order alternative that complements traditional rough-path methods.

Abstract

We develop an operator-theoretic framework for stochastic calculus with respect to rough fractional Brownian motion with Hurst parameter H < 1/2. Building on a covariant derivative defined via kernel factorization, we construct a closed unbounded operator on L2(Omega) adapted to the non-semimartingale setting. This approach yields explicit derivative representations for square-integrable functionals and provides a unified analytical framework compatible with rough path techniques. The results extend classical stochastic calculus beyond the semimartingale regime.

Stochastic Calculus for Rough Fractional Brownian Motion via Operator Factorization

TL;DR

This work develops an operator-theoretic stochastic calculus for rough fractional Brownian motion with by building an operator-covariant derivative as the adjoint of a divergence and introducing a predictable projection in the Cameron–Martin (RKHS) space. A central result is the factorization , which yields explicit first-order derivative formulas for functionals and a controlled martingale expansion that mirrors rough-path structure. The predictable component is identified with the Gubinelli derivative, linking the operator framework to rough-path theory while avoiding the construction of iterated integrals. The approach extends naturally to mixed semimartingale–rough processes, enabling unified stochastic calculus for models combining Brownian and rough components, with concrete applications to optimal filtering, rough-volatility hedging, and SDEs driven by rough noise, along with practical numerical schemes based on RKHS truncation. This framework provides conceptual clarity and computational advantages by grounding rough-stochastic calculus in Gaussian energy geometry and Malliavin-type adjointness, offering a first-order alternative that complements traditional rough-path methods.

Abstract

We develop an operator-theoretic framework for stochastic calculus with respect to rough fractional Brownian motion with Hurst parameter H < 1/2. Building on a covariant derivative defined via kernel factorization, we construct a closed unbounded operator on L2(Omega) adapted to the non-semimartingale setting. This approach yields explicit derivative representations for square-integrable functionals and provides a unified analytical framework compatible with rough path techniques. The results extend classical stochastic calculus beyond the semimartingale regime.
Paper Structure (39 sections, 8 theorems, 99 equations)

This paper contains 39 sections, 8 theorems, 99 equations.

Table of Contents

  1. Introduction
  2. Existing approaches for $H < \frac{1}{2}$
  3. Our contribution
  4. Organization
  5. Gaussian Volterra Processes and Cameron--Martin Spaces
  6. Setup
  7. Cameron--Martin (reproducing kernel Hilbert) space
  8. Fractional Brownian motion
  9. The Operator-Covariant Derivative
  10. Divergence operator
  11. Definition via Riesz representation
  12. Operator Factorization
  13. Predictable projection
  14. Main factorization theorem
  15. Explicit Computation Formulas
  16. ...and 24 more sections

Key Result

Lemma 3.3

$D_X$ is a closed operator on $L^2(\Omega)$.

Theorems & Definitions (30)

  • Definition 2.1: Gaussian Volterra process
  • Definition 2.2: Cameron--Martin space
  • Example 2.3: Fractional Brownian motion
  • Remark 2.4
  • Remark 3.1: Non-isometry for $H < \frac{1}{2}$
  • Definition 3.2: Operator-covariant derivative
  • Lemma 3.3: Closedness
  • proof
  • Proposition 3.4: Agreement with Malliavin derivative
  • proof
  • ...and 20 more