Truncated fractional moments of stable laws

TL;DR

This paper derives formulas for truncated fractional moments of stable distributions using special functions, providing explicit expressions for cases where the stability parameter exceeds one.

Contribution

It introduces new explicit formulas for truncated fractional moments of stable laws involving special functions, extending previous results.

Findings

Explicit formulas for $E X_+^p$ of stable laws.

Special functions related to multivariate stable densities.

Results for $E(X-a)_+$ when $eta > 1$.

Abstract

Expressions are given for the truncated fractional moments $E X_+^p$ of a general stable law. These involve families of special functions that arose out of the study of multivariate stable densities and probabilities. As a particular case, an expression is given for $E(X-a)_+$ when $\alpha > 1$.