= Solution
The explicit <stochastic exponential> solutions satisfy
$$
S_T=\widetilde S_T
\exp\left\{-\frac12(1-\rho^2)Y_T
+\sqrt{1-\rho^2}\int_0^T\sqrt{v_t}\,dW_t^\perp\right\}.
$$
Conditionally 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 therefore
$$
G\bigl(\widetilde S_T,(1-\rho^2)Y_T\bigr).
$$
Taking expectations and using part b proves the result.
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