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

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.