Characterization of Semimartingale Functions of Diffusion Processes, with Applications in Stochastic Optimal Control

Let X be a multidimensional diffusion process with non-degenerate diffusion and let f (t,x) be a function of n+1 variables. We show that the transformed process $f(t, X_t)$ is an Ito process if and only if the function $f$ admits a generalized derivative at $x$ and a generalized $L$ operator. We shall use this characterization to relate the BSDE for the value process of an optimal control problem with the Bellman equation for the value function of the same problem, when the controlled process is a Markov diffusion.