Calculation of $\ln\left( \frac{S_{1}(t)}{S_{2}(t)}\right)$ where $S$ are stocks Assume we have a probability space $(\Omega,\mathcal{F},\mathbb{P})$ where $\mathcal{F} =(\mathcal{F}_t)_{0 \leq t \leq \tau}$ is a Filtration of an incomplete finance market with stocks $S_j(t)$ for $j=1,2$. Their price process is supposed to be modeled by an Itô-process, driven by a Brownian Motion $\widehat{W} = (\widehat{W}^1, \widehat{W}^2)$, with the following solutions:
The price process $S_j(t)$ satisfy some SDE which solutions are given with respect to $\mathbb{P}$ by, where $i,j=1,2$:
$\begin{align}
S_{j}(t) & = e^{(r-\frac{1}{2} || \sigma_{j} ||^2 + \sigma_{i} \sigma_{j})\, t + \sigma_{j} \widehat{W}_t }
\end{align}$
where $r,\sigma_j,\sigma_i$ are constants. Now one can make the following calculations:
\begin{align}
P \left( S_{1}(s) \leq S_{2}(s) \Big\vert \mathcal{F}_{t}^{W} \right)
& =P \left( \frac{S_{1}(s)}{S_{2}(s)}  \leq  1 \Big\vert \mathcal{F}_{t}^{W} \right)\\
& = P \left( \ln\left( \frac{S_{1}(s)}{S_{2}(s)}\right)  \leq 0 \Big\vert \mathcal{F}_{t}^{W} \right)
\end{align}
So far so good. Now I'm at the point where i need help to calculate the last term. I got the solutions but I dont understand them. Any help is welcome. Those are the solutions:
Define $X := \ln\left(\frac{S_{1}(t)}{S_{2}(t)} \right)$. Under $\mathbb{P}$,
\begin{align}
X =&\ \ln\left( \frac{S_{1}(t)}{S_{2}(t)}\right) + \sum_{j=1}^{2} (\sigma_{1,j} -\sigma_{2,j} ) (\widehat{W}_s^{j}- \widehat{W}_t^{j}) \\
 & - \sum_{j=1}^{2} \left( \frac{1}{2} \left[\sigma_{1,j}^2-\sigma_{2,j}^2 \right] - \left[\sigma_{1,j} -\sigma_{2,j} \right] \sigma_{\pi(i),j} \right) (s-t)
\end{align} 
where $ s > t$ (why?). And $X$ is normal (why?) with mean $m(t)$ and variance $v(t)$, where:
\begin{align}
m(t) := \ln\left( \frac{S_{1}(t)}{S_{2}(t)}\right) - \sum_{j=1}^{2} \left( \frac{1}{2} \left[\sigma_{1,j}^2-\sigma_{2,j}^2 \right] - \left[\sigma_{1,j} -\sigma_{2,j} \right] \sigma_{\pi(i),j} \right) (s-t)
\end{align} 
and
\begin{align}
v(t)^2 := \sum_{j=1}^{2} (\sigma_{1,j} -\sigma_{2,j} ) (s-t)
\end{align}.
Also a why? Thanks for any explanations, help and tips.
FrakChris
 A: We assume that
\begin{align*}
d\left(\!
\begin{array}{c}
S^1(t)\\
S^2(t)
\end{array}
\!\right)
=\textrm{diag}\left(S^1(t), S^2(t)\right)\bigg[\left(\!
\begin{array}{c}
r\\
r
\end{array}
\!\right)dt +
\left(\!
\begin{array}{cc}
\sigma_{1,1} &\sigma_{1,2}\\
\sigma_{2,1} &\sigma_{2,2}
\end{array}
\!\right)d
\left(\!
\begin{array}{c}
W_t^1\\
W_t^2
\end{array}
\!\right)\bigg],
\end{align*}
where $\{W_t^1, t \ge0\}$ and $\{W_t^1, t \ge0\}$ are two independent standard Brownian motions. That is, for $i=1, 2$,
\begin{align*}
dS^i(t) = S^i(t)\bigg(rdt + \sum_{j=1}^2 \sigma_{i, j} dW_t^j \bigg).
\end{align*}
Then, for $s>t$,
\begin{align*}
S^i(s) = S^i(t)e^{\left(r-\frac{1}{2}\sum_{j=1}^2\sigma_{i,j}^2 \right)(s-t) + \sum_{j=1}^2\sigma_{i, j} \big(W_s^j-W_t^j \big)}.
\end{align*}
Moreover,
\begin{align*}
\ln \frac{S^1(s)}{S^2(s)} &= \ln \frac{S^1(t)}{S^2(t)}-\frac{1}{2}\sum_{j=1}^2\left(\sigma_{1,j}^2 -\sigma_{2,j}^2\right)(s-t) + \sum_{j=1}^2\left(\sigma_{1, j} -\sigma_{2, j}\right)\big(W_s^j-W_t^j \big)\\
&=\ln \frac{S^1(t)}{S^2(t)}-\frac{1}{2}\sum_{j=1}^2\left(\sigma_{1,j}^2 -\sigma_{2,j}^2\right)(s-t) \\
&\qquad+ \sqrt{\sum_{j=1}^2\left(\sigma_{1, j} -\sigma_{2, j}\right)^2 (s-t)}\,\frac{\sum_{j=1}^2\left(\sigma_{1, j} -\sigma_{2, j}\right)\big(W_s^j-W_t^j \big)}{\sqrt{\sum_{j=1}^2\left(\sigma_{1, j} -\sigma_{2, j}\right)^2(s-t)}}\\
&=\ln \frac{S^1(t)}{S^2(t)}-\frac{1}{2}\sum_{j=1}^2\left(\sigma_{1,j}^2 -\sigma_{2,j}^2\right)(s-t) + \sqrt{\sum_{j=1}^2\left(\sigma_{1, j} -\sigma_{2, j}\right)^2(s-t)}\, Z\\
&=m(t)+v(t) \, Z,
\end{align*}
where $Z$ is a standard normal random variable that is independent of $\mathcal{F}_t$, 
\begin{align*}
m(t) = \ln \frac{S^1(t)}{S^2(t)}-\frac{1}{2}\sum_{j=1}^2\left(\sigma_{1,j}^2 -\sigma_{2,j}^2\right)(s-t), 
\end{align*}
and 
\begin{align*}
v(t) = \sqrt{\sum_{j=1}^2\left(\sigma_{1, j} -\sigma_{2, j}\right)^2(s-t)}.
\end{align*}
Therefore, 
\begin{align*}
P\left(\ln \frac{S^1(s)}{S^2(s)} \le 0 \mid \mathcal{F}_t \right) &= \Phi\left( -\frac{m(t)}{v(t)}\right),
\end{align*}
where $\Phi$ is the cumulative distribution function of a standard normal random variable, assuming that $v(t) \neq 0$.
