Calculus via regularizations in Banach spaces and Kolmogorov-type path-dependent equations

TL;DR

This paper explores stochastic calculus via regularizations in Banach spaces and applies it to Kolmogorov path-dependent equations, introducing strong-viscosity solutions as an alternative to classical solutions in infinite dimensions.

Contribution

It connects Banach space stochastic calculus with functional stochastic calculus and introduces strong-viscosity solutions for path-dependent equations.

Findings

Established links between Banach space calculus and functional stochastic calculus.

Defined strong-viscosity solutions as an alternative to classical solutions.

Applied regularization techniques to study solutions of path-dependent equations.

Abstract

The paper reminds the basic ideas of stochastic calculus via regularizations in Banach spaces and its applications to the study of strict solutions of Kolmogorov path dependent equations associated with "windows" of diffusion processes. One makes the link between the Banach space approach and the so called functional stochastic calculus. When no strict solutions are available one describes the notion of strong-viscosity solution which alternative (in infinite dimension) to the classical notion of viscosity solution.