Local times

Suppose $f$ is $C^{2}$ with compact support and

Show that $g$ is $C^{2}$ and $g^{\prime \prime }=2 f$.

Let $L_{t}^{y}$ be the jointly continuous local times of a Brownian motion $W$. Show

Show the null set can be taken to be independent of $y$. Thus there is no need to take a subsequence $\varepsilon_{n}$ to get almost sure convergence to $L_{t}^{y}$.

Let $W$ be a Brownian motion and fix $t$. Show that the function $x \rightarrow \int_{0}^{t} \; \mathbb {1}_{(-\infty , x]}\left(W_{s}\right) d s$ is continuous, a.s., but that the function $x \rightarrow \; \mathbb {1}_{(-\infty , x]}\left(W_{t}\right)$ is not continuous.

Let $\left\{ \mathcal{F}_{t}\right\}$ be a filtration satisfying the usual conditions. Suppose $W_{t}$ is a Brownian motion and $X_{t}=W_{t}+A_{t}$, where $X_{t} \geq 0$ for all $t$, a.s., and $A_{t}$ is an increasing continuous adapted process such that $A$ increases only at those times when $X_{t}=0$. Suppose also that $X_{t}^{\prime }=W_{t}+A_{t}^{\prime }$, where $X_{t}^{\prime } \geq 0$ for all $t$, a.s., and $A_{t}^{\prime }$ is an increasing continuous adapted process that increases only when $X_{t}^{\prime }=0$. Show that $X_{t}^{\prime }=X_{t}$ and $A_{t}=A_{t}^{\prime }$, a.s., for all $t \geq 0$.

Let $W$ be a Brownian motion and $L_{t}^{0}$ the local time at 0 . Since $L_{t}^{0}$ is increasing, for each $\omega$ there is a Lebesgue-Stieltjes measure $d L_{t}^{0}$. Show that the support of $d L_{t}^{0}$ is equal to $\left\{ t: W_{t}=0\right\}$.

Since Theorem 14.1(2) states that $L_{t}^{0}$ does not increase when $W_{t}$ is not equal to 0 , what you need to show is that with probability one, if $W_{u}(\omega )=0$ and $t<u<v$, then $L_{v}^{0}(\omega )>L_{t}^{0}(\omega )$.

Use Tanaka's formula to show that if $L_{t}^{y}$ is the local time of Brownian motion at level $y$, $a \leq x \leq y \leq b$, and $T=\inf \left\{ t>0: W_{t} \notin [a, b]\right\}$, then

If $L_{t}^{0}$ is the local time of a Brownian motion at 0 , show that $L_{a t}^{0}$ has the same law as $\sqrt{a} L_{t}^{0}$.

Let $W$ be a Brownian motion with local times $L_{t}^{y}$. Set $L_{t}^{*}=\sup_{y} L_{t}^{y}$. Let $p>0$. Prove that there exist constants $c_{1}, c_{2}$ such that if $T$ is any finite stopping time,

The constants $c_{1}, c_{2}$ can depend on $p$, but not on $T$.

Hint: Use Exercise 12.11.

This exercise defines the local time of a continuous martingale. If $M$ is a continuous martingale, then $M_{t}^{2}$ is a submartingale and so equals a martingale plus an increasing process. The increasing process $L_{t}^{0}$ is called the local time of $M$ at 0 .

This exercise is a complement to Exercise 7.8. Let $W$ be a Brownian motion and let us define $Z=\left\{ t \in [0,1]: W_{t}=0\right\}$, the zero set. Let $\varepsilon \in (0,1 / 2)$ and let $\delta >0$. Fix $\omega$ and let $\left\{ B_{i}\right\}$ be any countable covering of $Z(\omega )$ by closed intervals such that the interiors of the $B_{i}$ 's are pairwise disjoint and the length of each $B_{i}$ is less than or equal to $\delta$. We write $B_{i}=\left[a_{i}, b_{i}\right]$.

Let $\varepsilon >0$. Since $L^{0}$ has the same law of the maximum of Brownian motion, there exists a $c$ (depending on $\omega$ ) such that

for each $0 \leq s \leq t \leq 0$. Write

Show that this implies that the Hausdorff dimension of $Z$ is at least $1 / 2$.