First passage times

Let $W$ be a standard Wiener process and let $X(t) = \exp \left\{ i \theta W(t)+\frac{1}{2} \theta^{2} t\right\}$ where $i = \sqrt{-1}$. Show that $X$ is a martingale with respect to the filtration given by $\mathcal{F}_{t} = \sigma (\left\{ W(u): u \leq t\right\} )$.

Let $T$ be the (random) time at which a standard Wiener process $W$ hits the 'barrier' in space-time given by $y = a t+b$ where $a \leq 0$ and $b > 0$; that is, $T = \inf \left\{ t: W(t) = a t+b\right\}$. Use the result of Exercise (13.4.1) to show that the density function of $T$ has Laplace transform given by

You may assume that the conditions of the optional stopping theorem are satisfied.

Let $W$ be a standard Wiener process, and let $T$ be the time of the last zero of $W$ prior to time $t$. Show that $\mathbb {P}\left(T \leq u\right) = (2 / \pi ) \sin^{-1} \sqrt{u / t}$ for $0 \leq u \leq t$.

Let $X$ and $Y$ be independent standard Wiener processes, so that $(X, Y)$ is a planar Wiener process. Show that the value of $Y$ at the time of the first passage of $X$ to the level $x > 0$ has the Cauchy distribution with density $f(y) = x /\left[\pi \left(x^{2}+y^{2}\right)\right]$.

Let $D(t) = W(t)-a t$ where $W$ is a standard Wiener process and $a \leq 0$ is a drift parameter. Use the result of Exercise (13.4.2) to show that the density function of the first passage time by $D$ to the point $b > 0$ is

Show that the standard Wiener process $W(t)$ and its maximum process $M(t) = \sup \left\{ W(s): 0 \leq s \leq t\right\}$ satisfy

where $\Phi$ is the $N(0,1)$ distribution function.

Find the density function of $\sqrt{T(x)}$ where $T(x)$ is the first passage time of a standard Wiener process to the point $x$. Show that it is the same as that of $1 /\left|Z\right|$ for a suitable normally distributed random variable $Z$.

Show that $T(x)$ has the same distribution as $T(1) x^{2}$, and deduce that the process $\left\{ T(x): x \geq 0\right\}$ is self-similar. Hence or otherwise, show that $T(x)$ has a stable distribution with exponent $\frac{1}{2}$. [See Feller 1971, p. 174.]

Feller's diffusion approximation to the branching process, Example (13.3.12) revisited.

Let $X = \left\{ X(t): t \geq 0\right\}$ be the diffusion model of Example (13.3.12) with $X(0) = x > 0$, and with instantaneous mean and variance $a x$ and $b x$ where $a < 0$ and $b > 0$. Let $Y = \int_{0}^{T} X(u) d u$ be the integrated size of the accumulated population up to the extinction time $T = \inf \left\{ t \geq 0: X(t) = 0\right\}$. Show that the moment generating function $m(x, \psi ) = \mathbb {E}_{x}\left[e^{-\psi Y}\right]$ satisfies

Deduce that $Y$ has a 'first-passage' distribution of the type of Exercise (13.4.2), and specify which.