Define
$$
\Delta A_t=U_{t-1}-\mathbb E[U_t\mid\mathcal F_{t-1}],
\qquad
\Delta M_t=U_t-\mathbb E[U_t\mid\mathcal F_{t-1}]
$$
for $t\geq1$, with $A_0=M_0=0$. The supermartingale property makes $\Delta A_t\geq0$, and it is $\mathcal F_{t-1}$-measurable, so $A$ is previsible and nondecreasing. The increments of $M$ have conditional mean zero, so $M$ is a martingale. Finally,
$$
\Delta U_t=\Delta M_t-\Delta A_t,
$$
and summation gives the <Doob decomposition in discrete time>
$$
U_t=U_0+M_t-A_t.
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 211 / 4 / a / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2024/iii/paper-211/4/a/solution