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Ive found a statement of Girsarnovs theorem that looks as follows

"Every $P$-semimartingale is a $Q$ semimartingale, in particular if $M$ is a local martingale then $\hat{M}_{t}=M_{t}-D_{t}^{-1}[M,D]_{t}$ is a $Q$ local martingale. Let $A_{t}=D_{t}^{-1}[M,D]_{t}$

Lets say $X=M_{1}+B$ is the semimartingale decomposition w.r.t $P$, do we just add and remove the above $A$, to get $X=M_{1}-A+A+B$ to get a $Q$ semimartingale?

How is X written as a $Q$ semimartingale? Or how do we see that it is a $Q$ semimartingale?

Number4
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    This might give some partial insight, https://math.stackexchange.com/questions/1396183/help-with-proof-of-girsanov-theorem?rq=1, however I am not sure that $A$ is of finite variation – user123124 Sep 25 '20 at 08:20

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Since we also have that $[M,X]=[\hat{M},X]$ are the same for all $X$ both under $\mathbb{P}$ and $\mathbb{Q}$ we know that $M$ and $\hat{M}$ are the same up to a constant. Therefore $X=\hat{M}+B$ is the $\mathbb{Q}$-semimartingale form

user123124
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