Girsanov Drift Changes and the Brownian Motion Under a New Measure

Summary

The note explains a sign convention in the Cameron–Martin–Girsanov theorem. Under the stated change of probability measure, the original process is described as having drift +ct, while subtracting ct produces a process that is Brownian under the new measure. The apparent mismatch comes from confusing the drifted process with the transformed process whose drift has been removed.

The answer relates the correction term to the quadratic covariation in the theorem and presents the subtraction as the way to recover zero drift under the new measure. This is a conceptual clarification rather than a full derivation: it does not discuss the theorem’s assumptions, broader change-of-measure setup, or applications to pricing and modeling. The result depends on the convention used for the density and which measure is treated as old or new, so those details matter when applying the sign rule.

Key ideas

• Under the new measure, the original Brownian process is described as acquiring a drift.
• Subtracting the drift term yields a process that is Brownian under the new measure.
• The stated correction is connected to quadratic covariation in the theorem.
• The sign depends on the chosen measure-change convention and density.

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# Question about the Cameron-Martin-Girsanov (CMG) theorem


# Question about the Cameron-Martin-Girsanov (CMG) theorem












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 $\mathbb{+} ct$?

## Answer by byouness (score 3, accepted)

https://quant.stackexchange.com/a/39914

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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