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Suppose we are working on the Black and Scholes Framework. There are only two assets, the risk-less bank account and a stock. The discounted process is a GBM under the physical measure with drift term $\mu -r$. Using the Girsanov-Cameron-Martin (G-C-M) theorem we can find the risk neutral martingale measure. My question is:

How do we know that it is unique?

For instance, in incomplete market models, G-C-M theorem holds but for a set of different measures. Does the uniqueness come from the Radon-Nikodym derivative?

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Basically the argument is that we have arrow-debreu securities (instrument that pays 1 if you arrive in a certain state). In the absence of arbitrage the price of this arrow-debreu security should be the same under both measures. But the price of an arrow-debreu security is simply the probability of that event happening. Hence both measures must be the same.

The link below describes it much more elegantly and in continuous time:

Unique risk neutral measure for Brownian Motion

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