ROUGH PATHS AND SYMMETRIC-STRATONOVICH INTEGRALS DRIVEN BY SINGULAR COVARIANCE GAUSSIAN PROCESSES

Alberto Ohashi (1) , Francesco Russo (2, 3, 4)

1 UnB - Universidade de Brasilia [Brasília]
2 ENSTA Paris - École Nationale Supérieure de Techniques Avancées
3 UMA - Unité de Mathématiques Appliquées
4 OC - Optimisation et commande

We examine the relation between a stochastic version of the rough path integral with the symmetric-Stratonovich integral in the sense of regularization. Under mild regularity conditions in the sense of Malliavin calculus, we establish equality between stochastic rough path and symmetric-Stratonovich integrals driven by a class of Gaussian processes. As a by-product, we show that solutions of multi-dimensional rough differential equations driven by a large class of Gaussian rough paths they are actually solutions to Stratonovich stochastic differential equations. We obtain almost sure convergence rates of the first-order Stratonovich scheme to rough paths integrals in the sense of Gubinelli. In case the time-increment of the Malliavin derivative of the integrands is regular enough, the rates are essentially sharp. The framework applies to a large class of Gaussian processes whose the second-order derivative of the covariance function is a sigma-finite non-positive measure on ${\mathbb R}^2$ + off diagonal.

Mots clés: Rough paths; Stratonovich integrals

Domaines: Probabilités [math.PR]

Alberto Ohashi, Francesco Russo. ROUGH PATHS AND SYMMETRIC-STRATONOVICH INTEGRALS DRIVEN BY SINGULAR COVARIANCE GAUSSIAN PROCESSES. Bernoulli, 2024, 30 (2), pp.1197-1230. ⟨10.3150/23-BEJ1629.short⟩. ⟨hal-03694046v2⟩