# $P(S_T > S_u \mid S_v = s_*)$

Let $u < v < T$ and assume $S_t$ follows a lognormal $((\mu - \sigma^2/2)t, \sigma^2 t)$ process. I'm interested in computing the conditional probability $$P(S_T > S_u \mid S_v = s_*)$$ where $s_* \in (0, \infty)$ is some known value (as is $S_u$). The idea is that we've already generated the value $s_*$ and, based on this, I'd like to compute the probability the stock is above some value at time $u < v$. My thought is that I can simply write $$P(S_T > S_u \mid S_v = s_*) = P(s_* e^{(\mu - \sigma^2/2)(T-v) + \sigma\sqrt{T - v}Z}> S_u)$$ but I'm not confident about the derivation.

Even better, I'd like to work out $$P(S_T \in B \mid \mathcal{F}_v)$$ for general $B \in \mathcal{B}$, but I thought I'd start with a concrete probability first.

If you assume that $(S_t)_{t\geq0}$ is a GBM with constant drift $\mu$ and volatility $\sigma$, then conditionally on $\mathcal{F}_v$ you can indeed write that: $$S_T = s_* \exp\left({\left(\mu - \frac{\sigma^2}{2}\right)(T-v) + \sigma\sqrt{T - v}Z} \right)$$ for any $T \geq v \geq 0$ provided $Z \sim N(0,1)$ and $S_v = s_*$ a.s.
Now assuming that $\mathcal{B}$ is a compact set $\mathcal{B}:=[a,b] \in \Bbb{R}$, then $$P(S_T \in \mathcal{B} \,\vert\, \mathcal{F}_v) = P(S_T < b \,\vert\, \mathcal{F}_v) - P(S_T < a \,\vert\, \mathcal{F}_v)$$