On It\^{o}'s formula for symmetric $\alpha $-stable L\'{e}vy process of index $1<\alpha\leq 2 $

TL;DR

This paper develops a generalized Itô's formula for symmetric -stable Lévy processes with index between 1 and 2, using Young integration and bounded variation theories, and introduces approximation schemes for related area integrals.

Contribution

It introduces a novel approach to Itô's formula for -stable Lévy processes using Young integration and bounded variation methods, extending existing stochastic calculus tools.

Findings

Established integration of functions w.r.t. local time for -stable processes

Derived a generalized Itô's formula for these processes

Proposed approximation schemes for area integrals

Abstract

We use Young integration (resp, bounded $p,q$-variation theory introduced in \cite{Feng-Zhao}) to establish integration of determinate functions with respect to local time of symmetric $\alpha$-stable L\'evy process, for $\alpha \in ]1,2]$, in one parameter case (resp, in two parameter case). We then apply these integrals to write the corresponding generalized It\^{o} formula. Furthermore, some approximations schemes of the area integral w.r.t local time are given.