An extension of the functional Ito formula under a family of non-dominated measures

Motivated by questions arising in financial mathematics, Dupire introduced a notion of smoothness for functionals of paths (different from the usual Fr'echet–Gat'eaux derivatives) and arrived at a generalization of It=o’s formula applicable to functionals which have a pathwise continuous dependence on the trajectories of the underlying process. We study nonlinear functionals which do not have such pathwise continuity and further work simultaneously under the family of continuous semimartingale measures on path-space. We do this without introducing a second component, as carried out by Cont–Fournie but by using old work of Bichteler which allows to keep a pathwise picture even for complex functionals