Math489/889
Stochastic Processes and
Advanced Mathematical Finance
Homework 9
Steve Dunbar
Due Mon, November 22, 2010
Let ${W}_{1}\left(t\right)$
be a Brownian motion and ${W}_{2}\left(t\right)$
be another independent Brownian motion, and $\rho $
is a constant between $-1$
and $1$.
Then consider the process $X\left(t\right)=\rho {W}_{1}\left(t\right)+\sqrt{1-{\rho}^{2}}{W}_{2}\left(t\right)$.
Is this $X\left(t\right)$
a Brownian motion?
Differentiate the c.d.f. of ${T}_{a}$
to obtain the expression for the p.d.f of ${T}_{a}$.
Show that $E\left[{T}_{a}\right]=\infty $
for $a>0$.
(Hint: use the Integral Comparison Test, see any calculus book in
the section on Improper Integrals.)