Question $4$

Let $B(t)$ be a standard Brownian motion starting at $B(0)=0.$ Define the process $x(t)$ as:

$x(t)=B(t{)}^2-t$

$($a$)$ Show that $x(t)$ is a martingale with respect to the natural filtration $F_t$ generated by the Brow$-$

nian motion $B(t).$

APM$4802/$A$05/0/2024$

$($b$)$ Suppose the water level of a dam is modeled as $L(t)=B(t{)}^2-t,$ where $B(t)$ is a Brownian

motion. What is the expected time until the dam's water level first reaches zero?

$($c$)$ Consider a particle moving in one dimension whose position $x(t)$ is described by a standard

Brownian motion starting from $x(0)=x.$ Show that $x(t)$ is a Markov process.