Signature SDEs from an affine and polynomial perspective

TL;DR

This paper introduces a class of signature stochastic differential equations (SDEs) that are affine and polynomial, enabling explicit formulas for their transforms and full law characterization, with practical numerical applications.

Contribution

It demonstrates that signature SDEs form affine and polynomial processes, providing explicit formulas for transforms and full law characterization using tensor algebra Riccati and linear ODEs.

Findings

Explicit formulas for Fourier-Laplace transforms derived

Signature SDEs are shown to be affine and polynomial processes

Numerical examples illustrate practical applicability

Abstract

Signature stochastic differential equations (SDEs) constitute a large class of stochastic processes, here driven by Brownian motions, whose characteristics are linear maps of their own signature, i.e. of iterated integrals of the process with itself, and allow therefore for a generic path dependence. We show that their prolongation with the corresponding signature is an affine and polynomial process taking values in the set of group-like elements of the extended tensor algebra. By relying on the duality theory for affine or polynomial processes, we obtain explicit formulas in terms of converging power series for the Fourier-Laplace transform and the expected value of entire functions of the signature process' marginals. The coefficients of these power series are solutions of extended tensor algebra valued Riccati and linear ordinary differential equations (ODEs), respectively, whose vector fields can be expressed in terms of the characteristics of the corresponding SDEs. We thus construct a class of stochastic processes which is universal (in a sense specified in the introduction) within Ito-diffusions with path-dependent characteristics and allows for an explicit characterization of the Fourier-Laplace transform and hence the full law on path space. The practical applicability of this affine and polynomial approach is illustrated by several numerical examples.