Set-Indexed Itô Calculus Along Paths

Set-indexed stochastic analysis and set-indexed stochastic calculus are faced here with a new approach of dimension’s reduction. We introduce a new tool (main flow) in order to deal with one-parameter calculus in set-indexed framework. We prove an Itô formula for any Brownian functional where the Brownian component is not a martingale on the whole set of indices but induces such a martingale. As first extensions, we provide definitions of bracket and local time in set-indexed context.