The elementary discrete integration-by-parts identity is
$$
[X]^{(n)}_t=X_t^2-2M_t^{(n)}.
$$
By the supplied fact, $M^{(n)}\to M$ in $L^2$ of the uniform norm. Define
$$
[X]_t=X_t^2-2M_t.
$$
Then
$$
\mathbb E\sup_{t\geq0}|[X]^{(n)}_t-[X]_t|^2
=4\mathbb E\sup_{t\geq0}|M_t^{(n)}-M_t|^2\longrightarrow0,
$$
which proves i, while
$$
X^2-[X]=2M
$$
is an $L^2$-bounded martingale, proving ii.

Choose a subsequence converging uniformly almost surely. For $s<t$, all complete dyadic increments between $s$ and $t$ contribute nonnegative squares; only the two boundary increments can affect monotonicity, and they vanish uniformly by continuity of $X$. Passing to the limit gives $[X]_s\leq[X]_t$. Thus $[X]$ is nondecreasing and is the <quadratic variation> of $X$.

Solution

Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 202 / 2 / a / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2021/iii/paper-202/2/a/solution