Tanaka formula for SDEs driven by fractional Brownian motion

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.