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Rough Path Renormalization from Stratonovich to Itô for Fractional Brownian Motion

Zhongmin Qian, Xingcheng Xu

TL;DR

Problem: stochastic calculus based on semimartingales cannot handle fractional Brownian motion with $H\neq\tfrac{1}{2}$, necessitating a pathwise Itô-type theory that yields zero-mean integrals. Approach: develop a Itô-type fractional rough-path calculus using a universal lift for one-forms and integrand dependent lifts for RDEs within Lyons rough path theory, including Stratonovich–Itô relations and space-time extensions. Contributions: define the Itô fractional Brownian rough path lift, establish Itô formulas for homogeneous and inhomogeneous cases, extend to Itô integration for fOU and RDEs, and apply the framework to fractional Black-Scholes pricing and fOU parameter estimation with Girsanov-based arbitrage absence results. Findings: obtain zero-mean pathwise integrals, arbitrage-free pricing under restricted trading strategies, and consistent rough-path estimators for fOU, supported by Monte Carlo demonstrations. Significance: provides a rigorous, zero-mean, pathwise Itô calculus for memory-bearing $\text{fBM}$ that supports robust pricing, hedging, and inference in memory-driven systems.

Abstract

This paper develops an Itô-type fractional pathwise integration theory for fractional Brownian motion with Hurst parameters \( H \in (\frac{1}{3}, \frac{1}{2}] \), using the Lyons' rough path framework. This approach is designed to fill gaps in conventional stochastic calculus models that fail to account for temporal persistence prevalent in dynamic systems such as those found in economics, finance, and engineering. The pathwise-defined method not only meets the zero expectation criterion but also addresses the challenges of integrating non-semimartingale processes, which traditional Itô calculus cannot handle. We apply this theory to fractional Black-Scholes models and high-dimensional fractional Ornstein-Uhlenbeck processes, illustrating the advantages of this approach. Additionally, the paper discusses the generalization of Itô integrals to rough differential equations (RDE) driven by fBM, emphasizing the necessity of integrand-specific adaptations in the Itô rough path lift for stochastic modeling.

Rough Path Renormalization from Stratonovich to Itô for Fractional Brownian Motion

TL;DR

Problem: stochastic calculus based on semimartingales cannot handle fractional Brownian motion with , necessitating a pathwise Itô-type theory that yields zero-mean integrals. Approach: develop a Itô-type fractional rough-path calculus using a universal lift for one-forms and integrand dependent lifts for RDEs within Lyons rough path theory, including Stratonovich–Itô relations and space-time extensions. Contributions: define the Itô fractional Brownian rough path lift, establish Itô formulas for homogeneous and inhomogeneous cases, extend to Itô integration for fOU and RDEs, and apply the framework to fractional Black-Scholes pricing and fOU parameter estimation with Girsanov-based arbitrage absence results. Findings: obtain zero-mean pathwise integrals, arbitrage-free pricing under restricted trading strategies, and consistent rough-path estimators for fOU, supported by Monte Carlo demonstrations. Significance: provides a rigorous, zero-mean, pathwise Itô calculus for memory-bearing that supports robust pricing, hedging, and inference in memory-driven systems.

Abstract

This paper develops an Itô-type fractional pathwise integration theory for fractional Brownian motion with Hurst parameters , using the Lyons' rough path framework. This approach is designed to fill gaps in conventional stochastic calculus models that fail to account for temporal persistence prevalent in dynamic systems such as those found in economics, finance, and engineering. The pathwise-defined method not only meets the zero expectation criterion but also addresses the challenges of integrating non-semimartingale processes, which traditional Itô calculus cannot handle. We apply this theory to fractional Black-Scholes models and high-dimensional fractional Ornstein-Uhlenbeck processes, illustrating the advantages of this approach. Additionally, the paper discusses the generalization of Itô integrals to rough differential equations (RDE) driven by fBM, emphasizing the necessity of integrand-specific adaptations in the Itô rough path lift for stochastic modeling.

Paper Structure

This paper contains 37 sections, 15 theorems, 210 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries on rough paths and fractional Brownian motion
  3. Rough paths
  4. Fractional Brownian motion
  5. Itô Integration against fractional Brownian motion
  6. Itô integrals of one forms against fBM
  7. Relation between Stratonovich and Itô rough integrals (I)
  8. Relation between Stratonovich and Itô rough integrals (II)
  9. The Itô formula
  10. Homogeneous Itô type formula
  11. Inhomogeneous Itô formula
  12. Differential equations driven by rough paths
  13. Basics of differential equations
  14. Relation between differential equations driven by different rough paths
  15. Examples and applications
  16. ...and 22 more sections

Key Result

Proposition 1

(CQ02, Theorem 2; FH14, Theorem 10.4) Let $B^{H}=(B^{H,1},\cdots,B^{H,d})$ be a $d$-dimensional fBM with the Hurst parameter $H\in(\frac{1}{3},\frac{1}{2}]$. Then $B^{H}$, restricted to an finite interval $[0,T]$, lifts via (fBm-rough) to a geometric rough path $\mathbf{B}^{H}=(1,B^{H},\mathbb{B}^{H

Figures (1)

  • Figure 1: Price of European call at time $0$ for different exercise time $T$ and varying Hurst parameter $H=0.50, 0.45, 0.40, 0.35$, where we take $\sigma=0.50$, $K=3$, $X_0=2.5$, $r=0.05$ as an example.

Theorems & Definitions (27)

  • Proposition 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • proof
  • Remark 1
  • Lemma 1
  • Remark 2
  • Corollary 5
  • Theorem 6
  • ...and 17 more