Take the tour ×  Let $X_t$ be a strict local martingale and let
$S$ and $T$ be stopping times with $S \leq T$. Prove that
$[X]_S=[X]_T$ implies that $X_t$ is almost surely constant on
$[S, T]$, where $[X]_S$ and $[X]_T$ is the quadratic variation
of the stopped process.  share|improve this question add comment
1 Answer  For continuous local martingales $X_t$, it is known
(theorem 3) that $X_t$ and $[X]_t$ have the same intervals of
constancy almost surely, by conditioning over the event $[S,T]$
you get the result in that case.  Nevertheless and contrary to
what I said before editing the answer, I have a found simple
counterexample for general local martingale (i.e. with jumps).
Let's take the (local) martingale $X_t= N_t-t$, (where $N_t$ is
a Poisson process of intensity 1), and for $S=S_1$ the first
jump time of $X$ and $T=S+1$ (here $S_i$ is the $i$-th jump time
of $N_t$), then the event $A=\{\omega\in\Omega s.t. T<S_2\}$ has
strictly positive probability equal to $e^{-1}$ (it is widely
know that inter-ar