Path-dependent processes from signatures

TL;DR

The paper develops a unified framework to represent non‑Markovian path‑dependent stochastic processes, including Volterra and delay types, through explicit infinite series of the signature of the time‑augmented driving Brownian motion. By recasting the solution as $X_t = \langle \boldsymbol{\ell}_t, \widehat{\mathbb{W}}_t\rangle$ and solving a linear algebraic equation in the extended tensor algebra, it disentangles infinite‑dimensional Markovian structure and enables robust, tractable approximations via truncations and shuffle algebra. The authors establish Itô calculus for infinite linear combinations, derive moment and conditional moment representations, and provide rigorous convergence criteria within the $\mathcal{A}_h$ and $\mathcal{A}_{\exp}$ frameworks. They apply the method to linear Volterra and delay equations as well as Gaussian Volterra processes, including fractional Brownian motion, and illustrate accuracy through numerical experiments and conditional moment formulas, offering a practical tool for non‑Markovian stochastic modeling.

Abstract

We provide explicit series expansions to certain stochastic path-dependent integral equations in terms of the path signature of the time augmented driving Brownian motion. Our framework encompasses a large class of stochastic linear Volterra and delay equations and in particular the fractional Brownian motion with a Hurst index $H \in (0, 1)$. Our expressions allow to disentangle an infinite dimensional Markovian structure and open the door to straightforward and simple approximation schemes, that we illustrate numerically.

Figures (5)

• Figure 1: Trajectories of a shifted Riemann-Liouville fractional Brownian motion (black) against their truncated time-dependent linear representation \ref{['eq:linear-RL']}, i.e. $\left \langle \bm{\ell}_{t + \varepsilon}^\textnormal{RL}, \widehat{\mathbb{W}}_{t}^{\leq M} \right \rangle$, for several truncation orders $M$ and $\varepsilon=1/52$.
• Figure 2: Trajectories of a delay process (black) against their truncated time-independent linear representation \ref{['eq:lde']}, i.e. $\left \langle \bm{\ell}^\textnormal{DE}, \widehat{\mathbb{W}}_{t}^{\leq M} \right \rangle$, for several truncation orders $M$.
• Figure 3: Trajectories of the shifted fractional Volterra process (black) against their truncated time-independent linear multifactor approximation representation \ref{['eq:lvol']}, i.e. $\left \langle \bm{\ell}^\textnormal{VOL}_n, \widehat{\mathbb{W}}_{t}^{\leq M} \right \rangle$, for several truncation orders $M$ and $y=0.25, a_1=0.25, b_1=-1, a_2=0, b_2=1, \varepsilon=1/52, T=1/12, \tau=1/106, n=10$.
• Figure 4: $m^{th}$ unconditional, (a) and (b), and conditional, (c) and (d), moments of the shifted fractional Volterra process (black) against their truncated time-independent linear multifactor approximation representation \ref{['eq:lvol']} to the $m^{th}$ (shuffle) power, i.e. $\left \langle (\bm{\ell}^\textnormal{VOL}_n) ^{\mathrel{\sqcup \mkern -3mu \sqcup} m}, \widehat{\mathbb{W}}_{t}^{\leq M} \right \rangle$, for several truncation orders $M$ and $y=0.25, a_1=0.25, b_1=-1, a_2=-0.1, b_2=1, \alpha=0.6, \varepsilon=1/52, T=1/26, \tau=1/106, n=10$.
• Figure 5: $m^{th}$ unconditional, (a) and (b), and conditional (c) and (d) moments of the Delayed equation process \ref{['eq:sde-DE-exp']} to the $m^{th}$ (shuffle) power, i.e. $\left \langle (\bm{\ell}^\textnormal{DE}) ^{\mathrel{\sqcup \mkern -3mu \sqcup} m}, \widehat{\mathbb{W}}_{t}^{\leq M} \right \rangle$, for several truncation orders $M$ and $z=0.25, a_1=0.125, b_1=-0.5, c_1=-0.5, \alpha_1=1, a_2=0, b_2=0.5, c_2=0.5, \alpha_2=-1$.

Theorems & Definitions (94)

• Remark 2.1
• Definition 2.1: Shuffle product
• Remark 2.2
• Remark 2.3
• Proposition 2.2
• proof
• Proposition 2.3
• proof
• Definition 2.4: Signature
• Example 2.4