The Ray–Knight theorems

Suppose $f$ is a non-negative continuous function whose support $[a, b]$ is a subset of $(0,1)$. Show that there is a unique solution to the ordinary differential equation $F^{\prime \prime }(x)=2 F(x) f(x)$, $F(1)=1, F^{\prime }(1)=0$, that $F$ is everywhere positive, and $F$ is bounded on $[0, \infty )$.

Hint: Since $f$ is zero in $(b, \infty )$, then $F^{\prime \prime }$ is zero there, and hence is of the form $F(x)=A x+B$ for some $A$ and $B$ for $x \geq b$. Since $F^{\prime }(1)=0$, conclude that $A$ is 0 .

Suppose $X_{t}$ is a solution to the one-dimensional SDE

Suppose $\sigma$ and $b$ are bounded and continuous and $f$ is a bounded and continuous function. What ordinary differential equation must $H(x)$ satisfy (in terms of $\sigma , b$, and $f$ ) in order that

be a martingale?

Suppose $X_{t}$ is a solution to the one-dimensional SDE

Suppose $\sigma$ and $b$ are bounded and continuous and $f$ is a bounded and continuous function. What partial differential equation must $K(x, t))$ satisfy (in terms of $\sigma , b$, and $f$ ) in order that

be a martingale?

Let $W$ be a Brownian motion and $L_{t}^{y}$ the local times at level $y$. Prove that local times at a fixed time $t$ are not a Markov process. That is, let $t>0$ be fixed and show that $\left(L_{t}^{y}, y \geq 0\right)$ is not a Markov process in the variable $y$.

Let $S$ be the first time two-dimensional Brownian motion exits the unit ball and let $\psi (\lambda )=\mathbb {P}_{0}\left(S>\lambda \right)$. If $W$ is a one-dimensional Brownian motion with local times $L_{t}^{x}$ and $T=\inf \left\{ t>0: W_{t}=1\right\}$, find the distribution of $Y=\sup_{0 \leq x \leq 1} L_{T}^{x}$ in terms of $\psi$, i.e., write $\mathbb {P}\left(Y \leq \lambda \right)$ in terms of the function $\psi$.

Suppose $x \in (0,1)$. With $W$ and $T$ as in Exercise 26.5, find the distribution of $L_{T}^{x}$.

Let $W$ be a one-dimensional Brownian motion with local times $L_{t}^{x}$. Let $T_{r}=\inf \left\{ t>0: L_{t}^{0}=r\right\}$. The law of the process $x \rightarrow L_{T_{r}}^{x}$ can be described as follows: