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.

Solution

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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2024/iii/paper-211/3/c/solution