We introduce a Time-Varying Exponentially Fading Memory signature, which can be viewed as a mean-reverting extension of the classical path signature with a time-dependent rate of reversion. The previously proposed exponentially fading memory signature corresponds to the special case where this dependence is constant, leading to a stationary setting. We show that this construction preserves several fundamental algebraic and analytical properties of classical signatures, including an adapted Chen identity, linearization, path reconstruction capability, and a universal approximation property. In particular, when applied to Brownian motion augmented with time, it evolves as an inhomogeneous Ornstein–Uhlenbeck process. From a numerical perspective, this approach outperforms both the classical signature and the exponentially fading memory signature when learning inhomogeneous Ornstein–Uhlenbeck dynamics with periodically varying mean-reversion speeds. Finally, within a statistical arbitrage framework, we demonstrate that this method provides more accurate entry and exit signals, leading to improved decision-making in trading strategies.
Nlend et al. (Tue,) studied this question.