Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 211 / 3 / c / Solution

The explicit stochastic exponential solutions satisfyConditionally on the path generated by $W$, the last stochastic integral is a centered Gaussian random variable with variance $Y_T$, because $W^{\perp }$ is independent of $W$. The conditional expectation of $g(S_T)$ is thereforeTaking expectations and using part b proves the result.