For a <simple predictable process>
$$
H_s=\sum_{i=0}^{n-1}H_i\mathbf1_{(t_i,t_{i+1}]}(s),
\qquad H_i\in L^\infty(\mathcal F_{t_i}),
$$
define
$$
(H\mathbin\cdot M)_t
=\sum_{i=0}^{n-1}H_i
\bigl(M_{t\wedge t_{i+1}}-M_{t\wedge t_i}\bigr).
$$
Each summand is a bounded predictable multiple of a martingale increment, so conditional expectation proves that $H\mathbin\cdot M$ is a martingale. Orthogonality of disjoint martingale increments gives
$$
\mathbb E|(H\mathbin\cdot M)_\infty|^2
=\sum_i\mathbb E\!\left[
H_i^2(M_{t_{i+1}}-M_{t_i})^2\right]<\infty.
$$
It is therefore an $L^2$-bounded continuous martingale.

Solution

Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 202 / 1 / b / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2022/iii/paper-202/1/b/solution