(Doob-Meyer decomposition in discrete time). Let \(\left(M_{n}ight)_{n \in \mathbb{N}}\) be a discretetime submartingale with respect to a filtration \(\left(\mathcal{F}_{n}ight)_{n \in \mathbb{N}}\), with \(\mathcal{F}_{-1}=\{\emptyset, \Omega\}\).

a) Show that there exists two processes \(\left(N_{n}ight)_{n \in \mathbb{N}}\) and \(\left(A_{n}ight)_{n \in \mathbb{N}}\) such that i) \(\left(N_{n}ight)_{n \in \mathbb{N}}\) is a martingale with respect to \(\left(\mathcal{F}_{n}ight)_{n \in \mathbb{N}}\), ii) \(\left(A_{n}ight)_{n \in \mathbb{N}}\) is non-decreasing, i.e. \(A_{n} \leqslant A_{n+1}\) a.s., \(n \in \mathbb{N}\), iii) \(\left(A_{n}ight)_{n \in \mathbb{N}}\) is predictable in the sense that \(A_{n}\) is \(\mathcal{F}_{n-1}\)-measurable, \(n \in \mathbb{N}\), and iv) \(M_{n}=N_{n}+A_{n}, n \in \mathbb{N}\).

Let \(A_{0}:=0\),

\[A_{n+1}:=A_{n}+\mathbb{E}\left[M_{n+1}-M_{n} \mid \mathcal{F}_{n}ight], \quad n \geqslant 0\]

and define \(\left(N_{n}ight)_{n \in \mathbb{N}}\) in such a way that it satisfies the four required properties.

b) Show that for all bounded stopping times \(\sigma\) and \(\tau\) such that \(\sigma \leqslant \tau\) a.s., we have

\[\mathbb{E}\left[M_{\sigma}ight] \leqslant \mathbb{E}\left[M_{\tau}ight]\]

Use the Stopping Time Theorem 14.7 for martingales and (14.7).