Let $\{B_t, \mathcal{F}_t; t\ge 0\}$ be a standard, one-dimensional Brownian motion. Can we construct a random time $S$ such that $P[0\le S < \infty] = 1$ and $W_t = B_{S+t} - B_S$ is not a Brownian motion?

MathOverflow is a question and answer site for professional mathematicians. It only takes a minute to sign up.

Sign up to join this community
Anybody can ask a question

Anybody can answer

The best answers are voted up and rise to the top

$\begingroup$
$\endgroup$

5
Let $\{B_t, \mathcal{F}_t; t\ge 0\}$ be a standard, one-dimensional Brownian motion. Can we construct a random time $S$ such that $P[0\le S < \infty] = 1$ and $W_t = B_{S+t} - B_S$ is not a Brownian motion?

lasttime $t$ such that $B_t = -t$. Then $W_t > -t$ for every $t$, and hence $W_t$ is not a Brownian motion. $\endgroup$ – Mateusz Kwaśnicki Feb 20 '20 at 21:57