[Paper Review] Integration by parts formula and applications for SDE driven by fractional Brownian motion

Xiliang Fan|arXiv (Cornell University)|Jun 5, 2012

Stochastic processes and financial applications18 references4 citations

TL;DR

This paper establishes a Driver-type integration by parts formula for stochastic differential equations (SDEs) driven by fractional Brownian motion (fBm) using a novel coupling method. By deriving shift Harnack inequalities from this formula, the authors prove the absolute continuity of the solution's law with respect to Lebesgue measure, providing a non-Malliavin calculus route to density existence for non-Markovian, non-semimartingale noise.

ABSTRACT

By constructing a new family of successful couplings, the Driver-type integration by parts formula is established for the operator associated with stochastic differential equation driven by fractional Brownian motion. As applications, shift Harnack type inequalities are presented and then the absolute continuity of the solution is proved.

Motivation & Objective

• To develop a new coupling-based method for deriving integration by parts formulas in the context of SDEs driven by fractional Brownian motion.
• To overcome the limitations of Malliavin calculus in proving the existence of densities for solutions to SDEs with non-Markovian, non-semimartingale noise.
• To establish shift Harnack inequalities as a tool to deduce absolute continuity of the solution's law.
• To provide a non-Malliavin alternative for studying regularity and density properties of solutions to SDEs with fBm.

Proposed method

• Constructs a new family of successful couplings via a change of measure using a Girsanov-type transformation for fractional Brownian motion.
• Applies the Driver integration by parts formula framework to the coupled SDE system under the new probability measure.
• Derives the integration by parts formula by analyzing the martingale component and using the bounded variation part of the Girsanov density.
• Establishes shift Harnack inequalities by combining the integration by parts formula with Young's inequality and moment estimates on the quadratic variation of the martingale component.
• Uses the derived shift Harnack inequalities to prove that the law of the solution is absolutely continuous with respect to Lebesgue measure.
• Employs the representation of fBm via a Wiener process through the kernel operator $ K_H^* $, enabling the use of classical stochastic calculus tools.

Experimental results

Research questions

• RQ1 Can a Driver-type integration by parts formula be established for SDEs driven by fractional Brownian motion, which is neither Markovian nor a semimartingale?
• RQ2 How can shift Harnack inequalities be derived from such an integration by parts formula?
• RQ3 Can the absolute continuity of the solution's law be proven without relying on Malliavin calculus?
• RQ4 What role does the coupling method play in handling the non-Markovian nature of fractional Brownian motion?
• RQ5 What are the explicit dependence structures (in terms of time T and Hurst parameter H) in the resulting inequalities and their implications for regularity?

Key findings

• The paper successfully establishes a Driver-type integration by parts formula for SDEs driven by fractional Brownian motion using a novel coupling construction.
• Shift Harnack inequalities are derived, with explicit dependence on time T, the Hurst parameter H, and the shift vector y, showing exponential decay in the shift parameter.
• The constants in the Harnack inequalities depend on time T, the Hölder regularity of the drift and diffusion coefficients, and the Hurst parameter H.
• The shift Harnack inequalities imply that the law of the solution $ X_T $ is absolutely continuous with respect to Lebesgue measure.
• The absolute continuity result is proven without using Malliavin calculus, offering a new analytical pathway for density existence in non-semimartingale settings.
• The proof relies on the interplay between the coupling method, Girsanov's theorem for fBm, and moment estimates on the martingale component of the Girsanov density.