Change of variable formulas for non-anticipative functionals on path space

We derive a change of variable formula for non-anticipative functionals defined on the space of R(d)-valued right-continuous paths with left limits. The functionals are only required to possess certain directional derivatives, which may be computed pathwise. Our results lead to functional extensions of the Ito formula for a large class of stochastic processes, including semimartingales and Dirichlet processes. In particular, we show the stability of the class of semimartingales under certain functional transformations. (C) 2010 Elsevier Inc. All rights reserved.