# Girsanov Theorem and Probability Measures

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.

• 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? Jan 13, 2020 at 9:46