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

TL;DR

This paper develops a change of variable formula for non-anticipative functionals on path space, extending Ito calculus to broader classes of stochastic processes and demonstrating stability of semimartingales under functional transformations.

Contribution

It introduces a functional change of variable formula for non-anticipative functionals with directional derivatives, extending Ito's formula to new classes of stochastic processes.

Findings

Extended Ito formula to semimartingales and Dirichlet processes.

Proved stability of semimartingales under certain functional transformations.

Provided pathwise computation of directional derivatives.

Abstract

We derive a functional change of variable formula for {\it non-anticipative} functionals defined on the space of right continuous paths with left limits. The functional is 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. Keywords: change of variable formula, functional derivative, functional calculus, stochastic integral, stochastic calculus, quadratic variation, Ito formula, Dirichlet process, semimartingale, Wiener space, F\"ollmer integral, Ito integral, cadlag functions.