Math Problems Vault

Problem 25.1

Show weak existence holds under the hypotheses of Proposition 25.3.

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Problem 25.2

Show weak existence holds under the hypotheses of Proposition 25.4.

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Problem 25.3

Here is an example of an SDE where weak uniqueness does not hold. Suppose $W$ is a one-dimensional Brownian motion and $\alpha \in \left(0, \frac{1}{2}\right)$ . Let $\sigma (x)=1 \wedge \left|x\right|^{\alpha }$ . Find two solutions to

d X_{t}=\sigma \left(X_{t}\right) d W_{t}

that are not equal in law.

Hint: One is the solution that is identically zero. The other can be constructed by time changing a Brownian motion by the inverse of the increasing process

2 \int _{0}^{t}\left(1 \wedge \left|X_{s}\right|^{2 \alpha }\right)^{-1} d s

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Problem 25.4

(1)(2)

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(1)

Suppose $a_{s}$ and $b_{s}$ are bounded predictable processes with $a_{s}$ bounded below by a positive constant. Let $W$ be a one-dimensional Brownian motion. Suppose $Y$ is a one-dimensional semimartingale such that

d Y_{t}=a_{t} Y_{t} d W_{t}+b_{t} d t, \quad Y_{0}=0

Prove that if $t_{0}>0$ and $\varepsilon >0$ , there exists a constant $c>0$ depending only on $t_{0}, \varepsilon$ , and the bounds on $a_{s}$ and $b_{s}$ such that

\mathbb {P}\left(\sup _{s \leq t_{0}}\left|Y_{s}\right|<\varepsilon \right)>c

(2)

Now let $W$ be $d$ -dimensional Brownian motion, let $x \in \mathbb {R}^{d}$ , and let $\sigma$ be a $d \times d$ matrix-valued function that is bounded and such that $\sigma \sigma^{T}(x)$ is positive definite, uniformly in $x$ . That is, there exists $\Lambda >0$ such that for all $x$ ,

\sum _{i, j=1}^{d} y_{i} y_{j}\left(\sigma (x) \sigma ^{T}(x)\right)_{i j} \geq \Lambda \sum _{i=1}^{d} y_{i}^{2}, \quad \left(y_{1}, \ldots , y_{d}\right) \in \mathbb {R}^{d}

Let $b$ be a $d \times 1$ matrix-valued function that is bounded. Let $X$ be the solution to

d X_{t}=\sigma \left(X_{t}\right) d W_{t}+b\left(X_{t}\right) d t, \quad X_{0}=x

Use Itô's formula to find an equivalent expression for $\left|X_{t}-x\right|^{2}$ . Then use (1) to prove that if $t_{0}>0$ and $\varepsilon >0$ , there exists a constant $c>0$ not depending on $x$ such that

\mathbb {P}_{x}\left(\sup _{s \leq t_{0}}\left|X_{s}-x\right|<\varepsilon \right)>c

Problem 25.5

This is the support theorem for solutions to SDEs. Let $X, x, \varepsilon$ , and $t_{0}$ be as in (2) of Exercise 25.4. Suppose $\psi :\left[0, t_{0}\right] \rightarrow \mathbb {R}^{d}$ is a continuous function with $\psi (0)=x$ . Use the Girsanov theorem to prove that there exists $c>0$ such that

\mathbb {P}_{x}\left(\sup _{s \leq t_{0}}\left|X_{s}-\psi (s)\right|<\varepsilon \right)>c

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Problem 25.6

Suppose weak uniqueness holds for the one-dimensional stochastic differential equation

d X_{t}=\sigma \left(X_{t}\right) d W_{t}, \quad X_{0}=x \tag {25.8}

where $W$ is a one-dimensional Brownian motion. Suppose also that there exists a process $X^{\prime }$ that is adapted to the minimal augmented filtration of $W$ with $X_{0}^{\prime }=x$ and $d X_{t}^{\prime }=\sigma \left(X_{t}^{\prime }\right) d W_{t}$ . Prove that pathwise uniqueness holds for (25.8).

Hint: Show there exists a measurable map $F$ from $C[0, \infty ) \rightarrow C[0, \infty )$ such that $X^{\prime }=F(W)$ . If $X^{\prime \prime }$ is another solution to (25.8), then weak uniqueness shows that the laws of $\left(X^{\prime \prime }, W\right)$ and $\left(X^{\prime }, W\right)$ are equal, hence $X^{\prime \prime }=F(W)=X^{\prime }$ .

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