Math489/889 Stochastic Processes and Advanced Mathematical Finance Homework 9

Steve Dunbar

Due Mon, November 22, 2010

Let $W_{1} (t)$ be a Brownian motion and $W_{2} (t)$ be another independent Brownian motion, and $ρ$ is a constant between $- 1$ and $1$ . Then consider the process $X (t) = ρ W_{1} (t) + \sqrt{1 - ρ^{2}} W_{2} (t)$ . Is this $X (t)$ a Brownian motion?
1. Differentiate the c.d.f. of $T_{a}$ to obtain the expression for the p.d.f of $T_{a}$ .
2. Show that $E [T_{a}] = \infty$ for $a > 0$ . (Hint: use the Integral Comparison Test, see any calculus book in the section on Improper Integrals.)