Several versions of It\^{ o}'s formula have been obtained in the context of the functional stochastic calculus. Here, we revisit this topic in two ways. First, by defining a notion of derivative along a functional, we extend the setting of the (semimartingale) functional It\^{ o}'s formula and corresponding calculus. Second, for L\'{ e}vy processes, an optimal local-time based It\^{ o}'s formula is obtained. Some quick applications are then given.
Categories: math.PR