For a continuous local martingale $M$ with $M_0=0$, its <stochastic exponential> is
$$
\mathcal E(M)_t=\exp\!\left(M_t-\frac12[M]_t\right).
$$
It is the unique solution of the <stochastic differential equation> $dZ_t=Z_t\,dM_t$, $Z_0=1$, and is a nonnegative <local martingale>.

Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 202 / 5 / a / ii / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2024/iii/paper-202/5/a/ii/solution