= Solution
The two processes with the same distribution are
$$
\boxed{(|B_t|)_{t\geq0}\ \overset d=\ (Z_t)_{t\geq0}}.
$$
Indeed, $(M_t)_{t\geq0}$ has almost surely <monotone function>[nondecreasing] paths, whereas $|B_t|$ and $M_t-B_t$ do not, so $M$ cannot be one member of the equal pair. The stated conclusion then follows from the premise that two of the three processes have the same law. More directly, it is the first coordinate of the <Lévy identity for Brownian motion>; both $|B|$ and $M-B$ are <Reflected Brownian motion>.
Back to article page
