Math Problems Vault

Problem 39.1

Show that the existence of a weak solution to (39.1) is equivalent to the existence of a solution to the martingale problem for $\mathcal{L}$ .

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Problem 39.2

Suppose the $a_{i j}$ are Lipschitz functions in $x$ and the matrices $a(x)$ are positive definite, uniformly in $x$ ; see Exercise 25.4. Show that we can find matrices $\sigma (x)$ so that each $\sigma_{i j}$ is a Lipschitz function of $x$ and $a(x)=\sigma (x) \sigma^{T}(x)$ for each $x$ .

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Problem 39.3

If $X$ is a solution to (39.1), give formulas for $A_{t}$ and $M_{t}$ in terms of $\sigma$ and $b$ , where $M_{t}$ is a local martingale, $A_{t}$ is a process whose paths are locally of bounded variation, and $|X_{t}|=M_{t}+A_{t}$ .

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Problem 39.4

(1)(2)

Let $A \in (-1, \infty )$ and let $X$ be a solution to (39.1), where all the $b_{i}$ 's are equal to 0, $a=\sigma \sigma^{T}$ , and

a_{i j}(x)=\frac{\delta _{i j}+A x_{i} x_{j} /|x|^{2}}{1+A}

for $x \neq 0$ , where $\delta_{i j}$ is equal to 1 if $i=j$ and 0 otherwise. Let $a(0)$ be the identity matrix.

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(1)

Prove that the matrices $a(x)$ are uniformly elliptic.

(2)

Show that $|X_{t}|$ has the same law as a Bessel process of order

\frac{d+A}{1+A} .

Conclude that if $A$ is sufficiently close to $-1$ , then $X$ is transient, i.e., $\lim_{t \rightarrow \infty }|X_{t}|=\infty$ , a.s., while if $A$ is sufficiently large, there exist arbitrarily large times $t$ such that $X_{t}=0$ .

Problem 39.5

(1)(2)(3)

Suppose for each $n \geq 1$ , $a_{i j}^{n}(x)$ is symmetric in $i$ and $j$ , is continuous in $x$ , and the matrix whose $(i, j)$ th entry is $a_{i j}^{n}(x)$ is positive definite, uniformly in $x$ and $n$ . Let

\mathcal{L}^{n} f(x)=\sum _{i, j=1}^{d} a_{i j}^{n}(x) \frac{\partial ^{2} f}{\partial x_{i} \partial x_{j}}(x) \tag {39.17}

for $f \in C^{2}$ . Suppose $a_{i j}^{n}(x)$ converges to $a_{i j}(x)$ uniformly in $x$ as $n \rightarrow \infty$ , and define $\mathcal{L}$ analogously to (39.17). Fix $x_{0}$ and let $\mathbb {P}_{n}$ be a solution to the martingale problem for $\mathcal{L}^{n}$ started at $x_{0}$ .

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(1)

Prove that $\mathbb {P}_{n}$ converges weakly to a solution $\mathbb {P}$ to the martingale problem for $\mathcal{L}$ started at $x_{0}$ .

(2)

Prove that if the $a_{i j}$ are continuously differentiable functions of $x$ whose first partial derivatives are bounded, then there exists a solution to the martingale problem for $\mathcal{L}$ started at $x_{0}$ .

(3)

Prove that if the $a_{i j}$ are continuous functions of $x$ , then there exists a solution to the martingale problem for $\mathcal{L}$ started at $x_{0}$ .

Problem 39.6

Suppose $X$ is a solution to $d X_{t}=\sigma \left(X_{t}\right) d W_{t}$ , where $W$ is a $d$ -dimensional Brownian motion, $\sigma (x)$ is a $d \times d$ matrix-valued function that is bounded, and $\sigma^{T} \sigma$ is positive definite, uniformly in $x$ . Prove the following estimate for the time to leave a ball: there exist constants $c_{1}$ and $c_{2}$ not depending on $x_{0}$ such that

c_{1} r^{2} \leq \mathbb {E}_{x_{0}}\left[\tau _{B\left(x_{0}, r\right)}\right] \leq c_{2} r^{2}, \quad r>0,

where $\tau_{B\left(x_{0}, r\right)}=\inf \left\{ t>0: X_{t} \notin B\left(x_{0}, r\right)\right\}$ .

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