# Proving completeness of $\mathcal H^p$

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.

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.
• I think I'm missing something. Your answer argues that $$\| M^{(n)} - N \| = \left\| \sum_{k\ge n} \left( M^{(k+1)} -M^{(k)} \right)\right\| \le \sum_{k \ge n} \left\| M^{(k+1)} - M^{(k)} \right\|.$$ However, the triangle inequality is valid only for finite sums. So actually we would need $$\lim_{m\to\infty} \left\| \sum_{k = n}^m \left( M^{(k+1)} -M^{(k)} \right)\right\| = \left\| \sum_{k\ge n} \left( M^{(k+1)} -M^{(k)} \right)\right\|.$$ That is, we need to be able to take the limits inside the integral that defines the norm. How do you know this step is valid? – Theoretical Economist Sep 11 at 15:53
• @TheoreticalEconomist The $L^p$ triangle inequality is valid for infinite sums: this follows from the triangle inequality for real numbers (which is valid for infinite sums) and monotone convergence theorem. – Cave Johnson Sep 12 at 1:27