Pathwise Itô Calculus for Rough Paths and Rough PDEs with Path Dependent Coefficients

Path-dependent Itô formulas under (p,q)-variations

“…In fact, this approach has been recently studied by many authors in the context of path-dependent PDEs and path-dependent optimal stochastic control problems. We refer the reader to e.g [14,15,16,27,9,10,19,4,25] for a detailed account on this literature. In this case, the usual space-time derivative operators are replaced by the so-called horizontal and vertical derivative operators, given by ∇ h F and ∇ v F , respectively.…”

A weak version of path-dependent functional Itô calculus

“…Recently, a new branch of stochastic calculus has appeared, known as functional Itô calculus, which results to be an extension of classical Itô calculus to non-anticipative functionals depending on the whole path of a noise W and not only on its current value, see e.g Dupire [20], Cont and Fournié [8,9], Cosso and Russo [14,15], Peng and Song [44], Buckdahn, Ma, and Zhang [5], Keller and Zhang [29], Ohashi, Shamarova and Shamarov [40] and Oberhauser [39]. Inspired by Peng [42], the issue of providing a suitable definition of path-dependent PDEs has attracted a great interest, see e.g Peng and Wang [43], Ekren, Keller, Touzi and Zhang [21], Ekren, Touzi and Zhang [22,23], Ekren and Zhang [24], Cosso and Russo [16] and Flandoli and Zanco [25].…”