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The Cameron-Martin-Girsanov theorem, in a simplistic way, states that:

The probability measure $\mathbb{P}$ is induced by a Wiener process $W(t)$. There exists another process $X(t)$ under the same measure which is a defined as $W(t)-rt$. Then, using the the Radon-Nikodym derivative $\frac{\mathrm{d}\mathbb{P_2}}{\mathrm{d}\mathbb{P}}=\exp(\int_0^t rdW_h -\frac{1}{2}\int_0^tr^2 dh)$, there exists an equivalent measure $\mathbb{P_2}$ under which $X(t)$ is a driftless Wiener process and $W(t)$ is a Wiener process with a drift.

My understanding is that the probability measure $\mathbb{P}$ is defined (induced) via the density of $W(t)$.

i.e. we could write $\mathbb{P}(A):=\int^{a}_{-\infty}f_{W_t}(h)dh$ for any event $A:W(t)\leq a$.

Does it make sense to discuss probabilistic events associated with $X(t)$ under $\mathbb{P}$? Of course, for any event $B:X(t)\leq b$, we could write:

$\mathbb{P}(B):=\mathbb{P}(X(t)\leq b)=\mathbb{P}(W(t)-r\leq b)=\mathbb{P}(W(t)\leq b+r)=\int^{b-r}_{-\infty}f_{W_t}(h)dh$.

In other words, we can discuss probabilistic events related to $X(t)$ under $\mathbb{P}$ as long as we stay within the framework of the probability measure $\mathbb{P}$ and write probabilities in terms of the density of $W(t)$.

As soon as we write $\mathbb{P}(B):=\mathbb{P}(X(t)\leq b)=\int^{b}_{-\infty}f_{X_t}(h)dh$, aren't we technically introducing a new probability measure, induced by the density of $X(t)$ itself?

Surely, $X(t)$ can induce its own measure $\mathbb{P_3}$, via its own density as aluded to above. The Radon-Nikodym derivative to go from $\mathbb{P}$ to $\mathbb{P_3}$ would simply be $\frac{\mathrm{d}\mathbb{P_3}}{\mathrm{d}\mathbb{P}}=\exp(-\int_0^t rdW_h -\frac{1}{2}\int_0^tr^2 dh)$. (We would then say that under $\mathbb{P_3}$, $W(t)$ has drift $-rt$ whilst under $\mathbb{P}$ it is a driftless Weiner process. Effectively $W(t)$ under $\mathbb{P_3}$ has the density of $X(t)$).

My question is the following then: wouldn't it make more sense to state the CMG Theorem as either:

(1) If $\mathbb{P}$ is induced by a driftless Wiener process $W(t)$, then there exists another measure (let's call it $\mathbb{Q}$ for no ambiguity with the above discussion), under which $W(t)$ has a drift $-rt$ and $\mathbb{Q}$ is defined via $\frac{\mathrm{d}\mathbb{Q}}{\mathrm{d}\mathbb{P}}=\exp(-\int_0^t rdW_h -\frac{1}{2}\int_0^tr^2 dh)$

OR:

(2) If $\mathbb{P}$ is induced by a Wiener process with a drift: $X(t)=W(t)-rt$, then there exists $\mathbb{Q}$ under which $X(t)$ is a Standard (driftless) Wiener Process and $\mathbb{Q}$ is defined via $\frac{\mathrm{d}\mathbb{Q}}{\mathrm{d}\mathbb{P}}=\exp(rX(t) -\frac{1}{2}tr^2)$.

It doesn't make sense to me to say that under $\mathbb{P}$ there is a Wiener process and another process which is Wiener with a drift (and therefore has a different probability density) and then apply one Radon-Nikodym derivative to modify two densities of both these processes (because each density defines its own associated probability measure - no?). I believe that the Radon-Nikodym derivative is process specific: we define it as a function of a specific process and that Radon-Nikodym derivative then defines a new probability measure for that specific process. To conclude, I find it incorrect to state the CMG Theorem the way it is stated.

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  • $\begingroup$ Let me shorten my question: can it be that $W(t)$ induces $\mathbb{P}$ (and therefore $\mathbb{P}$ is defined by the density of $W(t)$): and then, you have $X(t):= W(t) - rt$ (and therefore $X(t)$ has a different density) under the same measure $\mathbb{P}$ ? Wouldn't $X(t)$ define its own measure? $\endgroup$ – Jan Stuller Jan 13 at 9:46
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The question has now been answered here: https://math.stackexchange.com/questions/3514491/girsanov-theorem-and-different-probability-measures

It turns out that one needs to be more formal with probability measure definitions, if one wants to get to the bottom of it...

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