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

Abstract

Let $X=(X_t{)}_{t\ge 0}$ be a one-dimensional Lévy process such that each $X_t$ has a $C_b^1$-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.

Accepted for publication in Theory of Probability and Mathematical Statistics