A <simple predictable process> has the form
$$
H_s=\sum_{i=0}^{n-1}\xi_i\mathbf1_{(t_i,t_{i+1}]}(s),
$$
where each bounded $\xi_i$ is $\mathcal F_{t_i}$-measurable. Define
$$
(H\mathbin\cdot B)_t
=\sum_i\xi_i(B_{t\wedge t_{i+1}}-B_{t\wedge t_i}).
$$
Independent centered Brownian increments show directly by conditioning that this is a martingale. The same conditional expansion, using $\mathbb E[(B_v-B_u)^2\mid\mathcal F_u]=v-u$, shows that
$$
\boxed{(H\mathbin\cdot B)_t^2-\int_0^tH_s^2ds
\text{ is a martingale}.}
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 202 / 5 / 1 / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2019/iii/paper-202/5/1/solution