L2-bounded continuous martingale (${\mathcal{M}}^2$)

A continuous martingale $M$ is L2-bounded when ${\sup }_{t\ge 0}\mathbb{E}|M_t{|}^2<\infty$. The L2 martingale convergence theorem gives a terminal value $M_{\infty }$ with convergence in $L^2$ and almost sure convergence. The resulting conditional expectation identity is $M_t=\mathbb{E}[M_{\infty }\mid {\mathcal{F}}_t]$. Boundedness on each separate finite horizon is a weaker condition. The subspace with $M_0=0$ is commonly denoted ${\mathcal{M}}_0^2$.

For a continuous local martingale starting at zero, finite expected terminal quadratic variation is equivalent to being an L2-bounded continuous martingale. Stop at bounded path and bracket levels and use $\mathbb{E}(M_t^{\tau }{)}^2=\mathbb{E}[M{]}_{t\wedge \tau }$. Uniform integrability removes the localization; Doob L2 maximal inequality justifies convergence of squares. The terminal identity is $\mathbb{E}{M}_{\infty }^2=\mathbb{E}[M{]}_{\infty }$. For nonzero initial values one must additionally assume $M_0\in L^2$.

For a L2-bounded continuous martingale, the terminal and maximal norms satisfyThe first inequality follows from almost sure convergence. Apply the Doob L2 maximal inequality on $[0,T]$ and then the monotone convergence theorem as $T\to \infty$ for the second. These are equivalent norms on the space modulo indistinguishability of stochastic processes; neither inequality needs a zero initial value.