First-passage time processes and subordination

Suppose $W_{t}=\left(W_{t}^{1}, W_{t}^{2}\right)$ is a two-dimensional Brownian motion started at 0 . Let $\tau_{s}=$ $\inf \left\{ t>0: W_{t}^{1}>s\right\}$. Prove that $W_{\tau_{t}}^{2}$ is a Lévy process and determine the Lévy measure.

Hint: Use scaling to make a guess.

Let $W$ be a one-dimensional Brownian motion and let $L^{0}$ be the local time at 0 . Let $T_{t}$ be the inverse of $L^{0}$, that is, $T_{t}=\inf \left\{ s: L_{s}^{0} \geq t\right\}$. Show $T_{t}$ is a Lévy process and determine the Lévy measure.

Hint: Use scaling to get started.

Let $W_{t}$ be a one-dimensional Brownian motion, $L_{t}^{y}$ the local time at level $y$, and $T_{t}$ the inverse local time at 0 , that is, $T_{t}=\inf \left\{ s: L_{s}^{0} \geq t\right\}$. Let $x>0$ be fixed. Prove that $L_{T_{t}}^{x}$ is a Lévy process.

A stable subordinator $X$ of order $\alpha \in (0,1)$ is a Lévy process whose characteristic function is given by (42.6), where $b=\sigma^{2}=0$ and $m(d x)=c \; \mathbb {1}_{(x>0)}|x|^{-\alpha -1} d x$. Suppose $X$ is a stable subordinator of index $\alpha$ and $W$ is a Brownian motion. Show that, up to a deterministic time change, the process $Z_{t}=W_{X_{t}}$ is a symmetric stable process of index $2 \alpha$.

Hint: Start by using scaling.

Set $\tau_{a} = \inf \left\{ t > 0: W(t) = a\right\}$, where $W = \left\{ W(t), t \geqslant 0\right\}$ is a Wiener process, $a \in \mathbb {R}$. Prove that $\tau_{a} \stackrel{\mathscr {D}}{ = } a^{2} \tau_{1}$, i.e. the distributions of the random variables $\tau_{a}$ and $a^{2} \tau_{1}$ coincide: $\operatorname {Law}\left(\tau_{a}\right) = \operatorname {Law}\left(a^{2} \tau_{1}\right)$.

For $x \geqslant 0$ let $\tau_{x} = \inf \left\{ t \geqslant 0: W(t) = x\right\}$, where $W = \left\{ W(t), t \geqslant 0\right\}$ is a Wiener process. Prove that the process $\left\{ \tau_{x}, x \geqslant 0\right\}$ is a process with independent increments (hence Markov, by Theorem 3).

Theorem 3. Let $X = \left\{ X_{t}, t \in T\right\}$ be a process with independent increments taking values in $\mathbb {R}^{d}$, $d \geqslant 1$. Then $X$ is a Markov process (with respect to the natural filtration).

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.]

Prove Theorem (8.6.7): the property of being a Lévy process is preserved under time change by an independent subordinator.

Let $Z$ be a continuous-time Markov chain, and let $T$ be an independent subordinator. Show that $X(t) = Z(T(t))$ defines a Markov chain.

Let $W$ be the standard Wiener process, and let $T$ be an independent stable subordinator with $\mathbb {E}\left[e^{-\theta T(t)}\right] = \exp \left(-t \theta^{a / 2}\right)$ where $a \in (0,2)$ and $\theta \geq 0$. Show that the time-changed process $Y(t) = W(T(t))$ is a symmetric Lévy process.

Let $\left(W_{t}, t \geqslant 0\right)$ be a Wiener process, and let $\tau_{x}=\min \left\{ t : W_{t}=x\right\}$ for $x \geqslant 0$. Prove that the process $\tau =\left(\tau_{x}, x \geqslant 0\right)$ is Markov.

Let $\tau (z)$ be the random variable defined in the previous problem. Prove that the convolution of the distributions of the random variables $\tau \left(z_{1}\right)$ and $\tau \left(z_{2}\right)$ coincides with the distribution of the random variable $\tau \left(z_{1}+z_{2}\right)$.

Show that the distribution of the random variable $\tau (z)$ defined in problem 10.137 coincides with the distribution of the random variable $z^{2} \tau (1)$.

Find the characteristic function of the random variable $\tau (1)$ defined in problem 10.137.

Let $\left(B_{t}\right)_{t \geqslant 0}$ be a $\mathrm{BM}^{1}$ and set $\tau_{a} = \inf \left\{ s \geqslant 0: B_{s} = a\right\}$ where $a \in \mathbb {R}$. Show that $\tau_{a} \sim \tau_{ - a}$ and $\tau_{a} \sim a^{2} \tau_{1}$.