= Solution
Work on a <filtered probability space> with the <usual conditions for a filtration>. A continuous <martingale> belongs to the class of <L2-bounded continuous martingales> when
$$
\boxed{\sup_{t\geq0}\mathbb E|M_t|^2<\infty.}
$$
This means a uniform bound over the entire time interval, not merely being <square-integrable> at each individual time. The <L2 martingale convergence theorem> gives a terminal variable $M_\infty$ and <almost sure convergence> and convergence in the <Lebesgue space> $L^2$; moreover $M_t=\mathbb E[M_\infty\mid\mathcal F_t]$ by <conditional expectation>.
The <predictable sigma-algebra> $\mathcal P$ on $\Omega\times[0,\infty)$ is the smallest <sigma-algebra> making every left-continuous <adapted process> measurable. Equivalently, it is generated by
$$
A\times\{0\},\quad A\in\mathcal F_0,
\qquad
A\times(s,t],\quad 0\leq s<t,\ A\in\mathcal F_s,
$$
where the sample coordinate comes first. A <previsible process> is precisely a process measurable for this <predictable sigma-algebra>. Values at time zero matter for its definition, although they make no contribution to integration against a continuous <quadratic variation> starting at zero.
Back to article page
