Signature Kernel and Schwinger-Dyson Kernel Equations as Two-Parameter Rough Differential Equations
This work provides a theoretical foundation for two-parameter rough differential equations, benefiting researchers in stochastic analysis and machine learning who use signature methods, but the contribution is incremental as it extends existing rough path theory to a two-parameter setting.
The paper develops a rough-path framework for two-parameter rough differential equations on rectangular and simplicial domains, motivated by signature kernel and Schwinger-Dyson kernel equations. It establishes well-posedness, stability, and a numerical scheme with complexity estimates, validated by experiments.
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.