Signature Kernel and Schwinger-Dyson Kernel Equations as Two-Parameter Rough Differential Equations

TL;DR

This paper introduces a two-parameter rough-path framework for differential equations related to signature and Schwinger-Dyson kernels, extending existing theories to rough signals and providing numerical methods.

Contribution

It develops a novel two-parameter rough integration theory and applies it to establish well-posedness and stability of signature and Schwinger-Dyson kernel equations for rough paths.

Findings

Established well-posedness and stability of the equations

Extended the Schwinger-Dyson equation to rough signals

Provided a numerical scheme with complexity analysis

Abstract

We develop a rough-path framework for two-parameter rough differential equations on rectangular and simplicial domains, motivated by the signature kernel and Schwinger--Dyson kernel equations. The theory is formulated in spaces of jointly controlled rough paths and is based on a robust two-parameter rough integration framework. In particular, we introduce a notion of rough integration over two-dimensional simplices at low regularity extending previous results in the literature. Within this setting, we show that the signature kernel equation arises naturally as a two-parameter rough differential equation and establish well-posedness and stability. We also extend the Schwinger--Dyson kernel equation, previously formulated for bounded-variation paths, to rough driving signals, proving existence and uniqueness in appropriate controlled rough path spaces. In the smooth rough path regime, we relate the resulting equations to PDE and integro-differential formulations. Finally, we derive and analyse a numerical scheme for the rough Schwinger--Dyson equation, including runtime and memory complexity estimates, and illustrate its performance with numerical experiments.