Consider a filtration $\mathcal{F}_t$ and let $Y$ be integrable on the same probability space. Then $M_t=\mathbb{E}[Y|\mathcal{F_t}]$ is a martingale.
In Chapter 3, Exercise 8, Øksendal asks us to show a converse statement (3.8b) leveraging a Corollary (C.7) to show that if an $\mathcal{F}_t$-martingale is uniformly $L^p(\mathbb{P})$ for some $p>1$ then there exists an integrable $Y$ for which $M_t=\mathbb{E}[Y|\mathcal{F_t}]$.
I'm finding that I require the additional condition that $M_t$ is continuous; am I missing something here, or is the book simply missing the condition?
By (C.7), with continuity, we have that an integrable $Y$ exists such that $M_t\mathbin{\overset{ L^1 }{\longrightarrow}}Y$. Then, notice that $\mathbb{E}[Y-M_t|\mathcal{F}_t]=0$ if $\mathbb{E}[1_H(Y-M_t)]=0$ is for all events $H\in \mathcal{F}_t$.
But $1_H(M_s-M_t)\mathbin{\overset{ L^1 }{\longrightarrow}}1_H({Y-M_t})$ as $s\rightarrow\infty$. So $\mathbb{E}[{1_H(Y-M_t)}]=\lim_{s\rightarrow\infty}\mathbb{E}[{1_H(M_s-M_t)}]$ where for $s\ge t$
$$
\mathbb{E}[1_H({M_s-M_t})]=\mathbb{E}[1_H\mathbb{E}[{M_s-M_t}|{\mathcal{F}_t}]]=0\,\,.
$$
The continuity assumption seems load-bearing. Without continuity, let $\epsilon$ be a Rademacher random variable, $M_t=\epsilon 1\{t\ge 1\}$, and an unnaturally large constant filtration $\mathcal{H}_t=\sigma(\epsilon)$, then $M_t=0\neq \epsilon=\mathbb{E}[{\epsilon}|{\mathcal{H}_{t}}]$ for any $t<1$. Is it possible to construct a counterexample with a natural filtration?
(3.8b) Problem
(C.7) Corollary
Best Answer
There is an important property of stochastic processes that seems to be missing from Okesndal's book, or at least if it is there I couldn't find it. Specifically, a sub-martingale $M$ with respect to a filtration $(\mathcal F_t)$ satisfying the usual conditions has a right-continuous modification if and only if $t \mapsto \mathbb{E}[M_t]$ is right-continuous. In particular, if $M$ is a martingale, then $M$ has a right-continuous modification.
By considering this modification and using Theorem C.6 instead of Corollary C.7, we can prove the statement in the problem without using any additional assumptions.