BPHZ Renormalization in Gaussian Rough Paths

TL;DR

This paper develops a BPHZ renormalization procedure tailored for rough paths, connecting it with regularity structures, and demonstrates its effectiveness in stochastic PDEs and financial models.

Contribution

It introduces a novel BPHZ renormalization method for rough paths, bridging rough path theory with regularity structures and extending applications to finance.

Findings

BPHZ renormalization can be adapted to rough path theory.

The renormalized model converges in the rough path setting.

Application to stochastic PDEs and finance shows practical effectiveness.

Abstract

We construct a procedure for Bogoliubov-Parasiuk-Hepp-Zimmermann (BPHZ) renormalization of a rough path in view of the relation between rough path theory and regularity structure. We also provide a plain expression of the BPHZ-renormalized model in a rough path. BPHZ renormalization plays a central role in the theory of singular stochastic partial differential equations and assures the convergence of the model in the regularity structure. Here we demonstrate that the renormalization is also effective in a rough path setting by considering its application to rough path theory and mathematical finance.