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Assume you have the standard financial market built up of a Brownian motion. I have seen some books say that an equivalent local martingale measure imples no arbitrage, and some say that an equivalent martingale measure implies no arbitrage. These statements are not contradictory one is just stricter than the other.

But I am wondering, why do some state it with the "local" and some not? Is there some difference here when you include "local"? Is there some meaning behind it, or do they just simplify when they exclude it?

My last qustion is: An equivalent martingale measure is ofcourse also a local one, but is the existence of a equivalent local martingale measure equivalent to an existence of a martingale measure? That is, does the existence of a local martingale measure imply the existence of an equivalent martingale measure?

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The examples provided by Sin in their article Complications with Stochastic Volatility Models might help to answer your questions.

I'm transcribing the abstract below:

We show a class of stochastic volatility price models for which the most natural candidates for martingale measures are only strictly local martingale measures, contrary to what it is usually assumed in the finance literature. We also show the existence of martingale measures, however, and give explicit examples.

And this technical article (No arbitrage in continuous financial markets, by Criens) covers general integral tests for the existence and non-existence of EMM and ELMM (e.g., Theorem 3.1).

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