A Meyer-It\^o Formula for Stable Processes via Fractional Calculus

TL;DR

This paper develops a Meyer-Itô formula for strictly stable processes using fractional calculus, generalizing existing results and providing new semimartingale decompositions for functions of stable processes.

Contribution

It introduces a Meyer-Itô theorem for stable processes via fractional derivatives, extending Tanaka's formula and semimartingale decompositions to asymmetric cases.

Findings

Derived the inverse of the generator using fractional integrals

Established a Meyer-Itô formula with local time for stable processes

Generalized semimartingale decompositions for functions of stable processes

Abstract

The infinitesimal generator of a one-dimensional strictly $\alpha$-stable process can be represented as a weighted sum of (right and left) Riemann-Liouville fractional derivatives of order $\alpha$ and one obtains the fractional Laplacian in the case of symmetric stable processes. Using this relationship, we compute the inverse of the infinitesimal generator on Lizorkin space, from which we can recover the potential if $\alpha \in (0,1)$ and the recurrent potential if $\alpha \in (1,2)$. The inverse of the infinitesimal generator is expressed in terms of a linear combination of (right and left) Riemann-Liouville fractional integrals of order $\alpha$. One can then state a class of functions that give semimartingales when applied to strictly stable processes and state a Meyer-It\^o theorem with a non-zero (occupational) local time term, providing a generalization of the Tanaka formula given by Tsukada (2019). This result is used to find a Doob-Meyer (or semimartingale) decomposition for $|X_t - x|^{\gamma}$ with $X$ a recurrent strictly stable process of index $\alpha$ and $\gamma\in (\alpha-1,\alpha)$, generalizing the work of Engelbert and Kurenok (2019) to the asymmetric case.