Consider stochastic process $\{V_t\}$ that follows a Geometric Brownian motion $$dV_t = \mu V_t dt + \sigma V_t dZ_t$$ where $Z_t$ is a standard Brownian motion.
Consider the process $\{P_t\}$ where $P_t$ is the probability that the process $\{V_t\}$ hits a constant barrier $V_b$ between $t$ and $t + dt$. Is it possible to derive the dynamics of $P_t$?