It\^o's formula for finite variation L\'evy processes: The case of non-smooth functions

TL;DR

This paper extends Itô's formula to multi-dimensional finite variation Lévy processes for continuous functions with weak derivatives, broadening its applicability in theory and finance.

Contribution

It introduces a version of Itô's formula for multi-dimensional finite variation Lévy processes with non-smooth functions, expanding existing theoretical frameworks.

Findings

Extended Itô's formula to multi-dimensional finite variation Lévy processes.

Applicable to continuous functions with weak derivatives.

Potential applications in financial modeling.

Abstract

Extending It\^o's formula to non-smooth functions is important both in theory and applications. One of the fairly general extensions of the formula, known as Meyer-It\^o, applies to one dimensional semimartingales and convex functions. There are also satisfactory generalizations of It\^o's formula for diffusion processes where the Meyer-It\^o assumptions are weakened even further. We study a version of It\^o's formula for multi-dimensional finite variation L\'evy processes assuming that the underlying function is continuous and admits weak derivatives. We also discuss some applications of this extension, particularly in finance.