Consider a continuous $\mathbb{P}$ - semimartingale X which can be decomposed as M+A (M is local martingale and A is bounded variation process). Is it possible to change measure to $\mathbb{Q}$ s.t. $\mathbb{Q} << \mathbb{P}$ and X will be $\mathbb{Q}$ local martingale? I was trying to use Girsanov change of measure. However, I was not able to show existence of a local martingale $N$ whose cross variation with $M$ as $A$. Any help/reference is highly appreciated.
2026-04-07 16:17:35.1775578655
Transforming semimartingale to local martingale by change of measure
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Hi this is not always possible and it is the subject of one famous theorem in financial mathematics known as the Fundamental Theorem of Asset Pricing that claims that under some conditions there exist such measure change that turns (morally) semimartingales into local martingales.
In a series of articles by Delbean and Schachermayer (available on their personal webpages) they develop and prove the exact conditions under which this is true. It is VERY technical and not (IMO) suitable to develop the arguments here.
Best regards