Tanaka formula for SDEs driven by fractional Brownian motion

TL;DR

This paper derives a new Tanaka-type formula for solutions of SDEs driven by fractional Brownian motion with Hurst parameter greater than 1/2, revealing a complex structure involving Skorokhod integrals and Malliavin derivatives.

Contribution

It introduces the first Tanaka formula for non-linear SDEs driven by fractional Brownian motion, incorporating novel correction terms and a new convergence method.

Findings

The formula includes a local time represented by a double integral involving the Dirac delta and Malliavin derivative.

It applies to a broad class of SDEs with regularity assumptions and extends to convex functionals.

Special cases recover known identities for fractional Brownian motion and fractional Ornstein-Uhlenbeck processes.

Abstract

We derive a Tanaka-type formula for the solution of a stochastic differential equation (SDE) driven by fractional Brownian motion (fBm) with Hurst parameter $H > \frac{1}{2}$. While Tanaka formulas for the fractional Brownian motion itself have been established, a corresponding result for non-linear SDEs driven by fBm has so far been unavailable. Our formula reveals a structure not previously observed: it features both a Skorokhod integral and a Malliavin trace correction, where the analogue of the local time appears through a double integral involving the Dirac distribution and the Malliavin derivative of the solution. A second double integral captures the variation of the diffusion coefficient along the flow. A key step in our analysis is a novel method to establish $L^2$-convergence of the trace term, which avoids the use of white noise calculus and instead exploits Gaussian-type density estimates for the law of the solution. The result applies to a broad class of equations under suitable regularity assumptions and extends naturally to convex functionals. As special cases, we recover known identities for the fractional Brownian motion and the fractional Ornstein--Uhlenbeck process.