Let $\tau_n$ localize the nonnegative local martingale $M$. For $s\leq t$,
$$
\mathbb E(M_{t\wedge\tau_n}\mid\mathcal F_s)=M_{s\wedge\tau_n}.
$$
<Conditional Fatou lemma> and nonnegativity give
$$
\mathbb E(M_t\mid\mathcal F_s)
\leq\liminf_nM_{s\wedge\tau_n}=M_s.
$$
Thus $M$ is a <supermartingale>, recovering the general fact about a <nonnegative local martingale>.

Solution

Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 202 / 5 / b / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2022/iii/paper-202/5/b/solution