Math Problems Vault

Problem 27.1

Let $W$ be a Brownian motion, $x>0$ , and $T=\inf \left\{ t>0: W_{t}=x\right\}$ . If $L_{t}^{x}$ is the local time at $x$ , show that the distribution of $L_{T}^{x}$ is an exponential random variable. Determine the parameter of this exponential random variable.

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Problem 27.2

Let $W$ be a one-dimensional Brownian motion. This exercise asks you to prove that the normalized number of downcrossings by time $t$ converges to local time at 0 . If $a>0$ , let $S_{0}=0$ , $T_{0}=\inf \left\{ t: W_{t}=a\right\}$ , and for $i \geq 1$ ,

S_{i}=\inf \left\{ t>T_{i-1}: W_{t}=0\right\} , \quad T_{i}=\inf \left\{ t>S_{i}: W_{t}=a\right\}

Then $D_{t}(a)$ , the number of downcrossings up to time $t$ , is defined to be $\sup \left\{ k: S_{k} \leq t\right\}$ . Prove that there exists a constant $c$ such that

\lim _{a \rightarrow 0} a D_{t}(a)=c L_{t}^{0}, \quad \text{ a.s. }

where $L_{t}^{0}$ is local time at 0 of $W$ . Determine $c$ .

Hint: Use Exercise 18.5.

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Problem 27.3

(1)(2)

Let $\left(X_{t}, \mathbb {P}[x]\right)$ be a Brownian motion.

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(1)

Use the reflection principle to find

\mathbb {P}_{0}\left(X_{s}>-a \text{ for all } s \leq r\right) .

This is the same as $\mathbb {P}_{a}\left(T_{0}>r\right)$ , where $T_{0}$ is the first time the Brownian motion hits 0.

(2)

Let

A(a, r)=\left\{ f \in \mathcal{E}: \sup f(t)>a, \sigma _{f}>r\right\} , \quad B(r)=\left\{ f \in \mathcal{E}: \sigma _{f}>r, \sup f(t)>0\right\} ,

and

C(a)=\left\{ f \in \mathcal{E}: \sup f(t)>a\right\} .

Prove that

m(B(r))=\lim _{a \rightarrow 0} m(A(a, r))=\lim _{a \rightarrow 0}\left[m(C(a)) \times \mathbb {P}_{a}\left(T_{0}>r\right)\right]

and use this and (1) to compute $m(B(r))$ . By symmetry, $m\left(\left\{ f \in \mathcal{E}: \sigma_{f}>r\right\} \right)$ will be twice the value of $m(B(r))$ .

Remark.

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Let $\mathcal{E}$ be the collection of continuous functions $f$ with domain $[0, \infty )$ such that there exists a positive real $\sigma_{f}$ with $f(0)=0$ , $f\left(\sigma_{f}\right)=0$ , $f(t) \neq 0$ if $t \in \left(0, \sigma_{f}\right)$ , and $f(t)=0$ if $t>\sigma_{f}$ (furnished with the supremum norm). Given a Borel subset $A$ of $\mathcal{E}$ , the Brownian motion $W$ has had an excursion in $A$ by time $t$ if there exists a time $u$ and a function $f \in A$ such that $u+\sigma_{f} \leq t$ and $W_{u+s}(\omega )=f(s)$ for all $s \leq \sigma_{f}$ . Let $K_{t}(A)$ be the number of excursions of $W$ in $A$ by time $t$ . With $T_{r}=\inf \left\{ t>0: L_{t}^{0} \geq r\right\}$ and $N_{r}(A)=K_{T_{r}}(A)$ , the excursion measure is $m(A)=\mathbb {E}_{0}\left[N_{1}(A)\right]$ .

Problem 27.4

Let $W$ be a Brownian motion. Let $E_{t}(r)$ be the number of excursions of length larger than $r$ that have been completed by time $t$ . An excursion of length larger than $r$ means that $\sigma_{f}>r$ . Show that there exists a constant $c$ such that

\lim _{r \rightarrow 0} \sqrt{r} E_{t}(r)=c L_{t}^{0}, \quad \text{ a.s. }

Determine $c$ .

One interesting point here is that this shows that $L_{t}^{0}$ is determined entirely by the zero set $Z(\omega )=\left\{ t: W_{t}(\omega )=0\right\}$ .

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Problem 27.5

Let $\delta >0$ and $A_{\delta }=\left\{ f \in \mathcal{E}: \sup_{t}\left|f(t)\right|>\delta \right\}$ . Let $S_{1}=\inf \left\{ t: K_{t}\left(A_{\delta }\right)=1\right\}$ and $S_{2}=\inf \left\{ t>S_{1}: K_{t}\left(A_{\delta }\right)=2\right\}$ . Thus $S_{1}$ and $S_{2}$ are the times the first and second excursions in $A_{\delta }$ have been completed. Let $Y_{1}(t)$ be the excursion completed at time $S_{1}$ and define $Y_{2}(t)$ similarly. To be more precise, if $R_{1}=\sup \left\{ t<S_{1}: W_{t}=0\right\}$ , then $Y_{1}(s)=W_{R_{1}+s}$ if $s \leq S_{1}-R_{1}$ and $Y_{1}(s)$ is equal to 0 for all $s \geq S_{1}-R_{1}$ .

Prove that $Y_{1}$ and $Y_{2}$ are independent.

Hint: Use the strong Markov property at time $S_{1}$ .

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