A simple proof of functional Itô's lemma for semimartingales with an application

The Itô formula was extended recently by Dupire (2009) to functionals of paths of continuous semimartingales, and by Cont and Fournié (2010a) to functionals of paths of RCLL semimartingales. In contrast to the traditional formula that applies to functions of the current value of a process, these extensions apply to functionals of the history of a process. By modifying Dupire's setup we develop new proofs for both the continuous case and the more general RCLL case that are much simpler. We also examine an application to optimal control.