For which functions are $f(X_t)-\mathbb{E} f(X_t)$ and $g(X_t)/\mathbb{E} g(X_t)$ martingales?

TL;DR

This paper characterizes functions for which certain centered or normalized transformations of a Levy process are martingales, focusing on polynomially and exponentially bounded functions with specific moment conditions.

Contribution

It provides a complete characterization of functions making these transformations martingales for Levy processes with smooth densities and moment conditions.

Findings

Identifies all polynomially bounded functions f for which f(X_t)-E[f(X_t)] is a martingale.

Determines all exponentially bounded functions g for which g(X_t)/E[g(X_t)] is a martingale.

Establishes conditions on Levy processes and functions for martingale properties.

Abstract

Let $X=(X_t)_{t\geq 0}$ be a one-dimensional L\'evy process such that each $X_t$ has a $C^1_b$-density w.r.t. Lebesgue measure and certain polynomial or exponential moments. We characterize all polynomially bounded functions $f:\mathbb{R}\to\mathbb{R}$, and exponentially bounded functions $g:\mathbb{R}\to (0,\infty)$, such that $f(X_t)-\mathbb{E} f(X_t)$, resp. $g(X_t)/\mathbb{E} g(X_t)$, are martingales.