Generalized Ito formulae and space-time Lebesgue-Stieltjes integrals of local times

Generalised Ito formulae are proved for time dependent functions of continuous real valued semi-martingales.The conditions involve left space and time first derivatives, with the left space derivative required to have locally bounded 2-dimensional variation. In particular a class of functions with discontinuous first derivative is included. An estimate of Krylov allows further weakening of these conditions when the semi-martingale is a diffusion.