= Solution
For a continuous path $m$ of <finite variation> on $[0,t]$, every partition satisfies
$$
\sum_j(m_{t_{j+1}}-m_{t_j})^2\leq\max_j|m_{t_{j+1}}-m_{t_j}|\operatorname{Var}_{[0,t]}(m).
$$
Uniform continuity makes the right side tend to zero as the mesh tends to zero. The <quadratic variation> of the given <continuous local martingale> $M$ is therefore identically zero.
Stop $M$ when its magnitude first reaches an integer $n$, so the stopped process is a bounded, square-integrable <martingale>. The square identity, or the <Itô isometry>, gives $\mathbb E M_{t\wedge\tau_n}^2=\mathbb E[M]_{t\wedge\tau_n}=0$. Hence it is zero at each rational time almost surely, and continuity extends this simultaneously to all times. Let $n$ tend to infinity. This proves
$$
\boxed{M_t=0\text{ for every }t\geq0\text{ on one event of probability one}.}
$$
This is the <continuous finite-variation local martingale is constant> principle with initial value zero.
Back to article page
