Integrating the variance equation gives
$$
\int_0^T\sqrt{v_t}\,dW_t
=\frac1\gamma(v_T-v_0-\alpha T+\beta Y_T).
$$
The <Doléans-Dade exponential> solution of $d\widetilde S_t=\rho\widetilde S_t\sqrt{v_t}\,dW_t$ is
$$
\widetilde S_T
=S_0\exp\left\{
\rho\int_0^T\sqrt{v_t}\,dW_t
-\frac12\rho^2Y_T\right\}.
$$
Substitution yields
$$
\widetilde S_T
=S_0\exp\left\{
\frac\rho\gamma(v_T-v_0-\alpha T+\beta Y_T)
-\frac12\rho^2Y_T\right\}.
$$

Solution

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

 Solution

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