91Ó°ÊÓ

Let \(\mid X(t), t \geq 0]\) be Brownian motion with drift coefficient \(\mu\) and variance parameter \(\sigma^{2}\). That is, $$ X(t)=\sigma B(t)+\mu t $$ Let \(\mu>0\), and for a positive constant \(x\) let $$ \begin{aligned} T &=\operatorname{Min}[t: X(t)=x] \\ &=\operatorname{Min}\left\\{t: B(t)=\frac{x-\mu t}{\sigma}\right\\} \end{aligned} $$ That is, \(T\) is the first time the process \(\\{X(t), t \geq 0\\}\) hits \(x\). Use the Martingale stopping theorem to show that $$ E[T]=x / \mu $$

Let \(X(t)=\sigma B(t)+\mu t\), and for given positive constants \(A\) and \(B\), let \(p\) denote the probability that \(\\{X(t), t \geq 0\) \\} hits \(A\) before it hits \(-B\). (a) Define the stopping time \(T\) to be the first time the process hits either \(A\) or \(-B\). Use this stopping time and the Martingale defined in Exercise 19 to show that $$ \left.E\left[\exp \mid c(X(T)-\mu T) / \sigma-c^{2} T / 2\right]\right]=1 $$ (b) Let \(c=-2 \mu / \sigma\), and show that $$ E[\exp (-2 \mu X(T) / \sigma]]=1 $$ (c) Use part (b) and the definition of \(T\) to find \(p\). Hint: What are the possible values of \(\exp \left(-2 \mu X(T) / \sigma^{2}\right) ?\)

Let \([Z(t), t \geq 0]\) denote a Brownian bridge process. Show that if $$ Y(t)=(t+1) Z(t /(t+1)) $$ then \(\\{Y(t), t \geq 0]\) is a standard Brownian motion process.

Let \(\\{N(t), t \geq 0\\}\) denote a Poisson process with rate \(\lambda\) and define \(Y(t)\) to be the time from \(t\) until the next Poisson event. (a) Argue that \(\\{Y(t), t \geq 0\\}\) is a stationary process. (b) Compute \operatorname{Cov} [ Y ( t ) , Y ( t + s ) ] .

Let \(\\{X(t),-\infty

Let \(Y_{1}\) and \(Y_{2}\) be independent unit normal random variables and for some constant \(w\) set $$ X(t)=Y_{1} \cos w t+Y_{2} \sin w t, \quad-\infty