Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 4 / 29L / c / Solution

For $s<t$,The increment is independent of the Brownian filtration at time $s$, and its exponential factor has mean one by the Gaussian moment-generating function. Hence the conditional expectation of the discounted time-$t$ price is the discounted time-$s$ price.