I would like to compute the probability of a Geometric Brownian motion exceeding a certain value somewhere in a given period. We define the process by \begin{align*} d S_t = \mu S_t dt + \sigma S_t dWt, \end{align*} and using Itô's lemma we find the following solution \begin{align*} S_t = S_0 e^{(\mu - \frac{1}{2} \sigma^2)t + \sigma W_t}, \end{align*} Where $W_t$ denotes a Wiener process. Now we want to compute the probability of $S_t$ exceeding $B>S_0$ for some $t\in(0,T]$. We may now write \begin{align*} \mathbb{P}\left(\exists t\in(0,T]: S_t \geq B \right) &= \mathbb{P}\left(\max_{0<t\leq T} S_t \geq B \right)\\ &= \mathbb{P}\left(\max_{0<t\leq T}S_0 e^{(\mu - \frac{1}{2}\sigma^2)t+\sigma W_t} \geq B \right)\\ &= \mathbb{P}\left(\max_{0<t\leq T}(\mu - \frac{1}{2}\sigma^2)t+\sigma W_t \geq \ln \frac{B}{S_0} \right) \end{align*} So far so good. But now I would like to apply the reflection principle. But this may only be applied to a Wiener process, which is not the case if we set the drift of our GBM to 0, that is $\mu=0$.
To work around this problem I figured that I should apply the Girsanov theorem to make this a pure Wiener process. The problem here is that I don't know how to apply it. This is what I tried:
First we set $\nu=\frac{\mu}{\sigma}-\frac{\sigma}{2}$, then Girsanov tells us that there exists a probability measure $\mathbb{Q}$ defined by \begin{equation*} \mathbb{Q}(\omega) = e^{-\nu W_t^\mathbb{P}-\frac{\nu^2}{2}t}\mathbb{P}(\omega), \end{equation*} (Which I didn't really understand, as it seems that the new measure depends on a random variable)
such that \begin{equation*} W_t^{\mathbb{Q}} = \nu t + W_t^\mathbb{P} \end{equation*} is a Wiener process under $\mathbb{Q}$. (Note that $W_t^\mathbb{P}$ denotes our original Wiener process)
Then we find that \begin{align*} dW_t^\mathbb{P} = dW_t^\mathbb{Q} - (\frac{\mu}{\sigma} - \frac{\sigma}{2})dt. \end{align*} Substituting this in or differential equation gives \begin{align*} dS_t &= \mu S_t dt + \sigma S_t(dW_t^\mathbb{Q}-(\frac{\mu}{\sigma} - \frac{\sigma}{2})dt)= \frac{\sigma^2}{2}S_tdt + \sigma S_t dW_t^\mathbb{Q}. \end{align*} And using Itô's lemma now gives the solution \begin{equation*} S_t = S_0e^{\sigma W_t^\mathbb{Q}}. \end{equation*} Thus under the $\mathbb{Q}$ measure we can apply the reflection principle: \begin{align*} \mathbb{Q}\left(\exists t\in(0,T]: S_t \geq B \right) &= \mathbb{Q}\left(\max_{0<t\leq T}\sigma W_t^\mathbb{Q} \geq \ln \frac{B}{S_0}\right)\\ &= 2\mathbb{Q}\left(\sigma W_T^\mathbb{Q} \geq \ln \frac{B}{S_0}\right). \end{align*} And finally we can translate this between $\mathbb{P}$ and $\mathbb{Q}$ to get \begin{align*} \mathbb{P}\left(\exists t\in(0,T]: S_t \geq B \right) &= e^{\nu W_t^\mathbb{P}+\frac{\nu^2}{2}t}\mathbb{Q}\left(\exists t\in(0,T]: S_t \geq B \right)\\ &= e^{\nu W_t^\mathbb{P}+\frac{\nu^2}{2}t}2\mathbb{Q}\left(\sigma W_T^\mathbb{Q} \geq \ln \frac{B}{S_0}\right)\\ &= 2\mathbb{P}\left(\sigma W_T^\mathbb{Q} \geq \ln \frac{B}{S_0}\right)\\ &= 2\mathbb{P}\left(W_T^\mathbb{P} \geq \ln \frac{\frac{B}{S_0}-(\mu-\frac{\sigma^2}{2})T}{\sigma}\right) \end{align*} But this is just the same as applying the reflection principle on $(\mu-\frac{\sigma^2}{2})t+\sigma W_t^\mathbb{P}$ which isn't allowed in general.
I know for sure that the last steps are incorrect because it would imply that you can use the reflection principle regardless, but I'm not sure if I have made mistakes along the way. I would really appreciate someone pointing out my mistake to me and showing me the correct way to implement the Girsanov theorem.
Thanks in advance.
*Edit: I would like to refer to this answered question for a closed formula answer. Geometric Brownian Motion Probability of hitting uper boundary However I must confess that I still don't understand what is happening here.
Also apparently this method is described in "Brownian motion and stochastic calculus" by Ioannis Karatzas, Steven E. Shreve according to another article I read *
