How to compute the conditional probability for a geometric Brownian process?

Somewhat embarrassingly I'm stuck with something very elementary.

I want to find the conditional probability of a stock movement (GBM):

$$\mathbb{P} \big( S_t \geq b \vert S_s \leq b) $$

for $ t > s$. My main problem is to determine what $\mathbb P(S_t \geq b, S_s \leq b\big)$ equals.

Answer

By a certain algebraic manipulation, what you need is the probability $P(W_t \ge a, W_s \le c)$, which can be computed as below: \begin{align*} P(W_t \ge a, W_s \le c) &= P(W_t-W_s \ge a-W_s, W_s \le c)\\ &=E\big(E\left(1_{W_t-W_s \ge a-W_s} 1_{W_s \le c} \mid W_s \right)\big)\\ &=E\big(1_{W_s \le c}E\left(1_{W_t-W_s \ge a-W_s} \mid W_s \right)\big)\\ &=E\Big(1_{W_s \le c}\Big[1-\Phi\Big(\frac{a-W_s}{\sqrt{t-s}}\Big)\Big]\Big)\\ &=\Phi\Big(\frac{c}{\sqrt{s}} \Big)-\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\frac{a}{\sqrt{s}}}\Phi\Big(\frac{a-\sqrt{s}x}{\sqrt{t-s}} \Big)e^{-\frac{x^2}{2}} dx, \end{align*} where $\Phi$ is the cumulative distribution function of a standard normal random variable.