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Strict Positivity of the Density Ensures Equivalent Measures

Article Quant Q&A · Author: Politeten
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Summary

The note clarifies why a probability measure constructed from a Radon–Nikodym density in Girsanov’s theorem can be equivalent to the original measure, rather than merely absolutely continuous with respect to it. Absolute continuity follows from defining the new measure through a density, but equivalence also requires the reverse absolute continuity.

The key condition given is that the density process is strictly positive. A strictly positive density rules out events that have positive probability under the original measure but zero probability under the new measure, supplying the reverse direction. The note states this condition but gives no proof or discussion of when it holds, such as integrability or martingale requirements for the density process. It is therefore a concise conceptual answer, not a full treatment of the conditions needed to apply Girsanov’s theorem.

Key ideas

  • A Radon–Nikodym density directly gives absolute continuity of the new measure with respect to the original measure.
  • Strict positivity of the density supplies the reverse absolute continuity needed for equivalence.
  • The note states the key condition but does not develop the theorem’s other applicability requirements.

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# Understanding Girsanov's theorem in Bjork's book


# Understanding Girsanov's theorem in Bjork's book












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

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 $P$?

## Answer by quasi (score 2)

https://quant.stackexchange.com/a/33757

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

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.