Tanaka formula for SDEs driven by fractional Brownian motion

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.