C0,1-Itô chain rules and generalized solutions of parabolic PDEs

In this paper, we first establish an Itô formula for a finite quadratic variation process [Formula: see text] expanding [Formula: see text] when [Formula: see text] is of class [Formula: see text] in space and is absolutely continuous in time. Second, via a Fukushima–Dirichlet decomposition we obtain an explicit chain rule for [Formula: see text], when [Formula: see text] is a continuous semimartingale and [Formula: see text] is a “quasi-strong solution” (in the sense of approximation of classical solutions) of a parabolic PDE. Those results are applied in a companion paper to establish a verification theorem in stochastic differential games.