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Within my lecture notes, the following definition of the CMG theorem is given:

Under the probability measure $\mathbb{\tilde{P}}$ with density $\gamma_T = \exp(cW_T - \frac{c^2}{2}T)$, the process $W_t$, $0 \leq t \leq T$ is a Wiener process with drift $+ct$, while the process $$ \tilde{W}_t := W_t - ct \hspace{10mm} (*) $$ is a new Wiener process.

My question is, should $(*)$ not be $$ \tilde{W}_t := W_t \mathbb{+} ct $$ since we are talking about a new Wiener process with drift $\mathbb{+} ct$?

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According to CMG theorem, if $W_t$ is a Wiener process under the old measure, then under the new measure $\tilde{W_t}$ is a Wiener process, where: $$\tilde{W_t} = W_t + \langle cW, W \rangle_t = W_t - ct$$

Both statements express this same idea:

  • Moving from the old measure to new one adds a drift $ct$, so if you take a Wiener process (under the old measure) and express it under the new measure, you will get a Wiener process with a drift term $ct$.

  • So, if you want to get back to a Wiener process (i.e. drift = 0) under the new measure, then you have to remove this $ct$ term.

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