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

{c}
{title2=$\mathcal E(X)$}
{wiki}

For a continuous semimartingale $X$, its Doléans-Dade exponential is
$$
\mathcal E(X)_t=\exp\!\left(X_t-X_0-\frac12[X]_t\right).
$$
It solves $dZ_t=Z_t\,dX_t$ with $Z_0=1$.

= Stochastic exponential
{synonym}


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 Incoming links: Doléans-Dade exponential