Ramification of Volterra-type rough paths. (English) Zbl 1515.60336

Bruned, Yvain; Katsetsiadis, Foivos

Electron. J. Probab. 28, Paper No. 7, 25 p. (2023).

The authors study stochastic Volterra equations of the form \[ u(t) = u_0 + \sum_{i=0}^d \int_0^t k(t,r) f_i(u_r) d q^i_r, \quad u_0\in\mathbb{R}^e, \] where \(f_i\) are sufficiently regular vector fields on \(\mathbb{R}^e\), \(q\) is an \(\alpha\)-Hölder continuous signal and \(k\) is a suitable kernel, allowed to be singular in the diagonal \(t=r\).
They develop a pathwise existence and solution theory, which includes signals of arbitrarily low regularity \(\alpha>0\), taking inspiration from rough paths and the previous contributions by F. A. Harang and S. Tindel [Stochastic Processes Appl. 142, 34–78 (2021; Zbl 1481.60204)] and F. A. Harang et al. [Stoch. Dyn. 23, No. 1, Article ID 2350002, 50 p. (2023; Zbl 1511.45001)] which only covered \(\alpha>1/4\).
The main idea is that, instead of the iterated integrals of a path, one now keeps track of iterated integral convolutions with the Volterra kernel. This leads to a general framework of branched rough paths of Volterra type, or simply Volterra paths, on which an operation corresponding to an integral convolution, called convolution product and denoted by \(\star\), is well defined and satisfies some relevant algebraic properties. Once this is done, one can introduce a notion of controlled path à la Gubinelli and solve the Volterra equation by a fixed point argument.

Reviewer: Lucio Galeati (Lausanne)

MSC:

60L20	Rough paths
60L30	Regularity structures
60L70	Algebraic structures and computation

Keywords:

rough paths; Volterra equations

Citations:

Zbl 1481.60204; Zbl 1511.45001

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