= L2-bounded continuous martingale
{c}
{title2=$\mathcal M^2$}
A continuous <martingale> $M$ is L2-bounded when $\sup_{t\geq0}\mathbb E|M_t|^2<\infty$. 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 $\mathcal M_0^2$.
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