ProbabilityTheoryStochasticProcesses The Hilbert space of continuous Martingales started from 0 and bounded in is given by: Misplaced &\mathbb{H}^2 :&= \{ M \space \text{is a continuous Martingale} \space | \space M_0 = 0, \sup_{t \geq 0} \mathbb{E}[M_t^2 ] < \infty\} \\ &= \{ M \space \text{is a CLM} \space | \space M_0 = 0, \mathbb{E}[\left\langle M \right\rangle_\infty] < \infty\} \end{aligned}$$ with the inner product given by $\left\langle M, N \right\rangle_{\mathbb{H}^2} := \mathbb{E}[\left\langle M, N \right\rangle_\infty] = \mathbb{E}[M_\infty N_\infty]$. Processes $M_t \in \mathbb{H}^2$ have the following properties: - $M_t$ is [[Uniformly Integrable]] - $M_t$ converges to $\lim_{t \to \infty} M_t = M_\infty$ in $L^1$: $\mathbb{E}[|M_\infty - M_t|] \to 0$ - $M_t$ is the conditional expectation of $M_\infty$ up to time $t$: $$M_t = \mathbb{E}[M_\infty \space | \space \mathcal{F}_t]$$ - $\mathbb{H}^2$ is closed under stopping the process at a stopping time $T$: $$M^T \in \mathbb{H}^2$$