# Local martingales in discrete time

## TL;DR

The paper presents a new proof demonstrating that any discrete-time local martingale under measure P can be viewed as a true martingale under an equivalent measure Q, with control over the Radon-Nikodym derivative.

## Contribution

It introduces a novel proof technique based on an adaptation of Chris Rogers' argument, ensuring the existence of an equivalent measure Q with specific bounds on its density.

## Key findings

- Existence of an equivalent measure Q making S a Q-martingale
- Q can be chosen with density bounded by 1+ε for any ε>0
- Provides a new proof approach for fundamental theorem of asset pricing in discrete time

## Abstract

For any discrete-time $P$--local martingale $S$ there exists a probability measure $Q \sim P$ such that $S$ is a $Q$--martingale. A new proof for this result is provided. The core idea relies on an appropriate modification of an argument by Chris Rogers, used to prove a version of the fundamental theorem of asset pricing in discrete time. This proof also yields that, for any $\varepsilon>0$, the measure $Q$ can be chosen so that $\frac{dQ}{dP} \leq 1+\varepsilon$.