Local signature-based expansions
Federico M. Bandi, Roberto Renò, Sara Svaluto-Ferro
TL;DR
This work develops a unified, automatable framework for local (in-time) expansions of stochastic processes and their conditional moments using the time-extended Itô signature. It provides a process expansion $X_t=\langle c,\widehat{\mathbb W}_t\rangle+\varepsilon_n(t)$ with $\varepsilon_n(t)=o(t^{n/2})$, and extends this to explicit, computable expansions for both regular and irregular moments, including the characteristic function, to arbitrary order. The framework generalizes to discontinuous drivers by introducing the ${\overline \star}$ product and an extended signature ${\overline{\mathbb Z}}$, enabling joint treatment of jumps and diffusion and yielding higher-order, algorithmic expansions. Collectively, the results offer a tractable, depth-wise view into deep process characteristics (e.g., volatility of volatility) and provide practically implementable tools for short-horizon inference and pricing in continuous-time econometrics and finance.
Abstract
We study the local (in time) expansion of a continuous-time process and its conditional moments, including the process' characteristic function. The expansions are conducted by using the properties of the (time-extended) Ito signature, a tractable basis composed of iterated integrals of the driving deterministic and stochastic signals: time, multiple correlated Brownian motions and multiple correlated compound Poisson processes. We show that these properties are conducive to automated expansions to any order with explicit coefficients and, therefore, to stochastic representations in which asymptotics can be conducted for a shrinking time (t to 0), as in the extant continuous-time econometrics literature, but, also, for a fixed time (such that t smaller than 1) with a diverging expansion order. The latter design opens up novel opportunities for identifying deep characteristics of the assumed process.