A discussion on Itȏ's formula for non-smooth functions Seminar

Abstract: Itȏ's formula is one of the fundamental results in the theory of stochastic calculus. In simple words, the formula extends the notion of differentials for deterministic functions to that of random processes. It has deep applications in stochastic differential equations and in Mathematical Finance. The most famous application of this theorem is the derivation of Black-Scholes option prices.

The same as differential of deterministic functions, the smoothness assumption of the underlying functions plays an important role in Itȏ's formula. From an applied point of view, specifically in finance, the extension of Itȏ's formula to non-smooth functions is crucial. As far as we know, the best is done when the underlying function is the difference of two convex functions. However, this extension normally involves complicated random processes such as local times which make it very difficult to use in applications.

We obtain a version of Itȏ's formula for finite variation Lévy processes that applies more flexible assumptions on the underlying functions. In particular, instead of classical strong differentiability, we use weak derivatives. Finally we discuss the possible applications and generalization of this extended version.