Extend linear signature representations to nonlinear path-dependent volatility models

Develop linear signature representations with explicit analytical coefficients for nonlinear path-dependent stochastic volatility models, specifically the rough Heston model and the rough Bergomi model, by expressing the volatility process and its Itô integral as infinite linear combinations of time-extended Brownian motion signatures. This aims to generalize the existing analytical signature-based framework—currently limited to linear path-dependent equations such as linear Volterra and delay models—to these nonlinear volatility dynamics.

Background

The paper leverages rough path theory and signature transforms to handle non-Markovian stochastic volatility in option pricing. Prior work established linear signature representations with analytical coefficients for specific linear path-dependent processes, enabling tractable analysis and PDE methods.

However, these analytical constructions are restricted to linear models (e.g., linear Volterra and delay equations, Gaussian Volterra processes). The authors explicitly note that extending such analytical signature-based representations to nonlinear path-dependent volatility models like rough Heston and rough Bergomi remains unresolved. In this work they propose deep linear and deep nonlinear signature approaches as practical methods, but the analytical extension itself is still open.