Put
$$
p=\frac{8-\kappa}{8}>0,
\qquad q=\frac{8-\kappa}{\kappa}>0,
$$
so $M_t=\Upsilon_t^{-p}S_t^q$. Before $\tau_\epsilon$, one has $\Upsilon_t\geq\epsilon$ and $0<S_t\leq1$. Therefore
$$
0\leq M_{t\wedge\tau_\epsilon}\leq\epsilon^{-p}.
$$
The supplied continuous local martingale is thus bounded after stopping, and a bounded local martingale is a true martingale. Hence
$$
\boxed{(M_{t\wedge\tau_\epsilon})_{t\geq0}
\text{ is a bounded martingale}.}
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 203 / 2 / c / i / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2019/iii/paper-203/2/c/i/solution