Efficient evaluation of expectations of functions of a stable L\'evy process and its extremum

TL;DR

This paper develops efficient numerical methods for evaluating expectations involving a stable Lévy process and its supremum, using integral representations and conformal acceleration techniques.

Contribution

It introduces integral representations for expectations of functions of a stable Lévy process and its supremum, along with efficient numerical procedures utilizing conformal acceleration.

Findings

Derived integral representations for expectations of stable Lévy processes and their supremum.

Developed efficient numerical algorithms using conformal acceleration and trapezoid rule.

Validated methods with applications to distribution functions and expectations.

Abstract

Integral representations for expectations of functions of a stable L\'evy process $X$ and its supremum $\bar X$ are derived. As examples, cumulative probability distribution functions (cpdf) of $X_T, \barX_T$, the joint cpdf of $X_T$ and $\barX_T$, and the expectation of $(\be X_T-\barX_T)_+$, $\be>1$, are considered, and efficient numerical procedures for cpdfs are developed. The most efficient numerical methods use the conformal acceleration technique and simplified trapezoid rule.