Past exam of the mathematics course of the University of Cambridge / 2020 / ii / Paper 3 / 29K / b

Write $\Phi$ for the standard normal cumulative distribution function. Since $B_t\le M_t$, for $y\ge 0$ and $x>y$,For $x\le y$, the Brownian reflection principle givesThe joint distribution function is therefore

On the interior of the support $b<m$, differentiating the joint distribution of Brownian motion and its running maximum gives the joint probability densityMake the change-of-variables formula for a probability density $y=m$ and $z=m-b$. Its inverse is $m=y$, $b=y-z$, and its Jacobian determinant has absolute value one. Since $2m-b=y+z$, for $y,z\ge 0$ this yields

The two processes with the same distribution areIndeed, $(M_t{)}_{t\ge 0}$ has almost surely 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.