# A remark on conditions that a diffusion in the natural scale is a martingale

## TL;DR

This paper investigates conditions under which a diffusion process in the natural scale behaves as a martingale, providing simple and analytic proofs for existing results.

## Contribution

It offers straightforward, analytic proofs for known conditions that ensure a diffusion in natural scale is a martingale.

## Key findings

- Identifies conditions for martingale property in diffusions
- Provides simplified proofs for existing results
- Enhances understanding of diffusion behavior in natural scale

## Abstract

We consider a diffusion processes $\{ X_t \}$ on an interval in the natural scale. Some results are known under which $\{ X_t \}$ is a martingale, and we give simple and analytic proofs for them.