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

Franziska Kühn; René L. Schilling

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

Probability 2021-10-19 v3

Authors: Franziska Kühn , René L. Schilling

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.

Keywords

levy process martingale polynomially bounded function exponentially bounded function density

Cite

@article{arxiv.2107.11974,
  title  = {For which functions are $f(X_t)-\mathbb{E} f(X_t)$ and $g(X_t)/\mathbb{E} g(X_t)$ martingales?},
  author = {Franziska Kühn and René L. Schilling},
  journal= {arXiv preprint arXiv:2107.11974},
  year   = {2021}
}

Comments

Accepted for publication in Theory of Probability and Mathematical Statistics

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

Abstract

Keywords

Cite

Comments

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