Let $\mathcal H^p$, with $p \in [1,\infty)$, be the space of all (continuous-time) martingales $M$ such that $$ \|M\|_{\mathcal H^p} := \mathbb E\left[\sup_t \left| M_t\right|^p\right]^{1/p} < \infty. $$
I want to show that (identifying indistinguishable martingales) $\mathcal H^p$ is complete. Could someone help me complete the following proof?
Suppose that $\left\{ X^n \right\}$ is a Cauchy sequence in $\mathcal H^p$, so that, for every $\varepsilon > 0$, there is an $N$ such that $$ \| X^n -X^m \|_{\mathcal H^p} < \varepsilon $$ for all $n,m \ge N$. Since $$ \sup_t \mathbb E\left[\left| X^n_t - X^m_t \right|^p\right] \le \| X^n -X^m \|_{\mathcal H^p}^p, $$ we have that $\left\{ X^n_t \right\}$ is uniformly (in $t$) Cauchy in $L^p$, and thus must converge uniformly in $L^p$ to some $X_t \in L^p$.
It is straightforward to verify that $X$ must be a martingale. How do you show that $X^n \to X$ in the $\mathcal H^p$-norm?
By passing to a subsequence if necessary, we can assume that $X^n_t \to X_t$ uniformly almost surely. That is, $$ \sup_t | X_t^n -X_t |^p \to 0 \quad \text{a.s.} $$ One might then be able to appeal to an appropriate convergence theorem to show that $$ \mathbb E \left[ \sup_t | X_t^n -X_t |^p \right] \to 0. $$ We can then use the fact that $\{X^n\}$ is Cauchy to recover convergence for the original sequence. Equivalently, one can show that $\sup_t | X_t^n -X_t |^p$ is uniformly integrable. Unfortunately, I don't know how to show this.
Another thought is to use Doob's $L^p$ inequality, but that only works for $p > 1$.
Any suggestions? I suspect this should be rather simple, but my analysis is quite rusty.
The proof is quite standard.
By Chebyshev and Borel-Cantelli one may find a subsequence $\{M^{(n)}\}$ such that $\sum_n(M^{(n)}_t-M^{(n-1)}_t)$ converges uniformly a.s.; denote its limits by $N_t$. It suffices to show $M^{(n)}\to N$ in $\mathcal H^p$, which amounts to prove $\sum_{n\ge n_0}\|M^{(n)}-M^{(n-1)}\|_{\mathcal H^p}$ converges to zero. By choosing the subsequence properly, e.g. in a way such that $\|M^{(n)}-M^{(n-1)}\|_{\mathcal H^p}<2^{-n}$, the conclusion is obvious.