M = (M$_{t}$)$_{t\ge 0}$ is a right-continuous/cadlag/continuous R$^{d}$-valued martingale with respect to a filtration F and pe[1,$\infty $). We suppose that E[||M$_{t}$||$^{p}$] < $\infty $ for all teR$_{+}$, that is M is a p-integrable martingale.
(a) τ is a [0,$\infty $]-valued F$^{+}$-stopping time. Show that $E[||M_{t}^{τ}||^{p}]\le E[||M_{t}||^{p}]$ for all teR$_{+}$ and that M is also a p-integrable martingale with the same continuity property as M.
(b) (τ$_{n}$)$_{neN}$ is an increasing sequence of [0,$\infty $]-valued F$^{+}$-stopping time and $\lim_{n\to \infty }$ τ$_{n}=\infty $. Show that $\lim_{n\to \infty }$ E[$\Vert M_{t}^{τ_{n}}\Vert ^{p}$]=E[$\Vert M_{t}\Vert ^{p}$]
My solution:
(a) Corollary:(M$_{t}$)$_{teT}$ is a right-continuous martingale with respect to the Filtration F and τ is F$^{+}$-stopping time. Then the stopped process M$^{τ}_{t}$:=M$_{τ\land t}$ has the same properties with respect to the filtration F.
Since τ is F$^{+}$-stopping time and M is a right-continuous p-integrable martingale, we can apply the corollary and we get that the stopped process M$_{τ\land t}$ is also a right-continuous p-integrable martingale.
Lemma: If M = (M$_{t})_{teT}$ taking values in a convex set $A \subset R^{d}$ and f:A->R a convex function such that Y$_{t}$ :=f(M$_{t}$) is integrable for every teT. Then (Y$_{t}$)$_{teT}$ is a submartingale.
Since p-norms are convex for $p \ge 1$, we define our function $f(M_{τ\land t}):= \Vert M_{τ \land t} \Vert ^{p}$ (which is integrable for every teT) and apply the theorem and get that $(Y_{t})_{teT}=(\Vert M_{τ\land t} \Vert ^{p})_{teT}$ is a submartingale.
$(\Vert M_{τ\land t} \Vert ^{p})_{teT}$ is a submartingale, so for $s \le t$ $\Vert M_{τ\land s}\Vert ^{p} \le E[\Vert M_{τ\land t} \Vert ^{p} |F_{s}]$
$E[\Vert M_{τ\land s} \Vert ^{p}] \le E[E[\Vert M_{τ\land t} \Vert ^{p}|F_{s}]]$ = $E[\Vert M_{τ\land t} \Vert ^{p}|F_{s}] $= $E[\Vert M_{τ}1_{τ \le t} \Vert ^{p}|F_{s}] $+ $E[\Vert M_{t} 1_{τ > t} \Vert ^{p}|F_{s}]$
That is about as far as I could come to the finally solution. Any hints or have you noticed any mistakes?
(b) From (a) and the condition given in the exercise (E[$\Vert M_{t} \Vert ^{p}] < \infty $), we have that $E[\Vert M_{τ \land t} \Vert ^{p}] \le E[\Vert M_{t} \Vert ^{p}]$<$ \infty$ , thus (M$_{τ∧t}$) is p-integrable.
$\lim_{n\to \infty}$ $E[\Vert M_{τ_{n} \land t} \Vert ^{p}]$=$\lim_{n\to \infty}$ $E[\Vert M_{τ_{n}} 1_{τ_{n} Dominated convergence theorem applies so: $\lim_{n\to \infty}$ $E[\Vert M_{τ_{n}}1_{τ_{n} By Fatou's lemma: lim inf $_{n->∞}$ $E[\Vert M_{t}1_{τ_{n} \ge t} \Vert ^{p}] \ge $E[lim $\inf_{n->∞}$ $\Vert M_{t}1_{τ_{n} \ge t} \Vert^{p}$]=E[$\Vert M_{t}1_{ \infty \ge t} \Vert ^{p}$] Is this correct? I am not sure how to come till the end of the proof. I would be more than thankful if someone could help me.