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In Bjork's arbitrage theory in continuous time, he writes


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S, essentially we define $Q$ using $h_t$, and then pick $h_t$ so $Q$ is a martingale measure.

But, $Q$ needs to be an equivalent measure. We know from Radon-Nikodym that when we define $Q$ using this method, $Q$ WILL be absolutely continuous with respect to $P$. That is, $Q << P$.

But that doesn't mean $P \sim Q.$ We also need $P <<Q$.

So why is the $Q$ as defined using $L_t$ and $h_t$ equivalent? The Radon-Nikodym theorem only gives tells us that it is absolutely continuous wrt $P$?

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The process $L$ is strictly positive. This implies that $Q$ is equivalent to $P$ and not merely absolutely continuous.

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