$C^{0,1}$ -It{ô} chain rules and generalized solutions of parabolic PDEs

In this paper we first establish an Itô formula for a finite quadratic variation process $X$ expanding $f(t,X_t),$ when $f$ is of class $C^2$ in space and is absolutely continuous in time. Second, via a Fukushima-Dirichlet decomposition we obtain an explicit chain rule for $f(t,X_t)$, when $X$ is a continuous semimartingale and $f$ is a ``quasi-strong solution’’ (in the sense of approximation of classical solutions) of a parabolic PDE.