A simple proof of a result of A. Novikov

TL;DR

This paper provides simple proofs for key exponential martingale results related to continuous local martingales, clarifying conditions under which certain exponential moments are finite and equal to one.

Contribution

The paper introduces straightforward proofs of Novikov's result, simplifying the understanding of exponential martingale conditions for continuous local martingales.

Findings

Proves finiteness of exponential moments under specific conditions.

Establishes the equality E[exp(M_∞ - <M>_∞/2)] = 1.

Simplifies existing proofs of Novikov's theorem.

Abstract

We give simple proofs that for a continuous local martingale M_t: 1) \liminf_{\epsilon->0} \epsilon \log Ee^{(1-\epsilon) <M>_\infty /2} < \infty ==> E\exp(M_\infty - <M>_\infty /2) = 1, 2) \liminf_{\epsilon->0} \epsilon \log\sup_{t>=0} Ee^{(1-\epsilon)M_t/2} < \infty ==> E\exp(M_\infty - <M>_\infty /2) = 1 .