We recognize that geometric Brownian motion is an Itô process. An Itô process is a stochastic process with stochastic differential of the form
$$dX_t = a_tdt + b_t dW_t$$
where $dW_t = W_{t+dt} \; – W_t$ and $W_t$ is a $\mathbb{P}$ Brownian motion.
Theorem: The solution of the Black and Scholes equation is given by the discounted expectation of the derivative’s payoff conditional on $\widehat{S}_t = S$.
$$V(S,t) = e^{-(T-t)}\mathbb{E}\biggl( F (\widehat{S}_T) \bigg{|} (\widehat{S}_t = S)\biggr)$$
This expectation solution is a direct consequence of re-interpreting the Black and Scholes PDE as a relation in which the underlying price process is idealised as $\widehat{S}_t$ such that it’s stochastic differential has zero drift. (Unlike the actual $S_t$ price process with non-zero drift. $\widehat{S}_t$ solves the SDE
$$d\widehat{S}_t = r\widehat{S}_t dt + \sigma \widehat{S}_t dW_t$$
assuming no dividends (dividend rate $q=0$).