Characteristic function of the positive part of a random variable and related results, with applications

TL;DR

This paper derives integral formulas for the characteristic function of the positive part of a random variable, providing new insights and applications in areas like stock options and random walks, including a refined version of Spitzer's identity.

Contribution

It introduces integral expressions for the characteristic function of the positive part of a random variable and applies these to improve results in stock options and random walk theory.

Findings

Derived integral formulas for the characteristic function of max(0,X)

Applied formulas to stock options and random walks

Obtained a more explicit form of Spitzer's identity

Abstract

Let $X$ be an arbitrary real-valued random variable (r.v.), with the characteristic function (c.f.) $f$. Integral expressions for the c.f.\ of the r.v.'s $\max(0,X)$ in terms of $f$ are given, as well as other related results. Applications to stock options and random walks are presented. In particular, a more explicit and compact form of Spitzer's identity is obtained.