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

Rama Cont (1) , David-Antoine Fournie (2)

1 LPMA - Laboratoire de Probabilités et Modèles Aléatoires
2 Dept of Mathematics, Columbia University

We derive a functional change of variable formula for 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: stochastic calculus; functional calculus; Ito formula; Malliavin derivative; semimartingale; Dirichlet process

Domains: Probability [math.PR]; Functional Analysis [math.FA]

Rama Cont, David-Antoine Fournie. Change of variable formulas for non-anticipative functionals on path space. Journal of Functional Analysis, 2010, 259 (4), pp.1043-1072. ⟨10.1016/j.jfa.2010.04.017⟩. ⟨hal-00471318v2⟩