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Fix $T \ge 0$ and let $\mathcal{R}_c^2$ be the set of continuous adapted stochastic processes $X$ such that $\|\sup_{t\leq T} |X_t| \|_{L^2} < \infty$ with metric $$d(X,Y) = \|\sup_{t \leq T} |X_t -Y_t|\|_{L^2}$$

I want to show that $(\mathcal{R}_c^2, d)$ is a complete metric space (sidenote for those interested: this is part of the proof of why SDEs with Lipschitz terms have a unique strong solution, and is just asserted in what I'm reading).

It is trivial to show that it's a metric space (given that you know $L^2$ is a metric space) so I just need help with the completeness.


My ideas:

Let $X^n$ be a $d$-Cauchy sequence so that for each $t \leq T$ we have $$E(X_t^n-X_t^m)^2 \leq E\left(\sup_{t \leq T} (X_t^n - X_t^m)^2\right) \xrightarrow{m,n \rightarrow \infty} 0$$ so that $X_t^n$ is Cauchy in $L^2$ and has a limit $X_t$.

Define $X \equiv (X_t)_{t \le T}$. We need to show that $E(\sup_{t \leq T} (X_t^n - X_t)^2) \rightarrow 0$ but I'm completely stuck here.

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1 Answer 1

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Someone please tell me if this is correct:

I proceed as in the standard proof that $L^2$ is complete. Choose a subsequence $\{n_k\}_{k \in \mathbb{N}}$ such that $Y^k \equiv X^{n_k}$ satisfies $d(Y^k, Y^{k+1}) < \frac{1}{2^k}$

Further define $H_n = \sum_{k=1}^{n-1} \sup_{t \leq T} |Y^{k+1}_t - Y^k_t|$.

We see that $H_\infty \equiv \sup_n H_n < \infty$ almost surely since, by the monotone convergence theorem and Minkowski's inequality:

$$\|H_\infty\|_{L^2}^2 = \sup_n \|H_n\|_{L^2}^2 \leq \sup_n \sum_{k=1}^{n-1}d(Y^k, Y^{k+1}) = \sup_n \sum_{k=1}^{n-1} \frac{1}{2^k} = \sum_{k=1}^{\infty} \frac{1}{2^k} = 1$$

Thus for $m,n$ large and $\epsilon > 0$, $$\textbf{(1)} \quad \quad \sup_{t \leq T} |Y^{m}_t - Y^n_t| \leq H_m-H_{n-1} < \epsilon $$

Thus, $Y^n$ is a Cauchy sequence of continuous functions on $[0,T]$ and this is a complete metric space w.r.t. the supremum metric, so that there exists a continuous limit $Y \equiv (Y_t)_{t \leq T}$

Now $\sup_{t \leq T} |Y^{n}_t - Y_t| \rightarrow 0$ and taking $m \rightarrow \infty$ in the first inequality in the first inequality in (1) has $\sup_{t \leq T} |Y^{n}_t - Y_t| \leq H_\infty \in L^2$ so that we may use the dominated convergence theorem to say that $Y^n \rightarrow Y$ in the metric space $(\mathcal{R}_c^2,d)$. The uniqueness of limits for Cauchy sequences concludes the result. $\quad \quad \square$

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