Past exam of the continuous martingales and stochastic calculus course of the University of Oxford / 2023 / 3

The optional stopping theorem for a right-continuous martingale $M$ on a filtered probability space satisfying the usual conditions for a filtration states that, for bounded stopping times $\sigma \le \tau$,For a submartingale the conditional equality becomes $\mathbb{E}[M_{\tau }\mid {\mathcal{F}}_{\sigma }]\ge M_{\sigma }$, and for a supermartingale it becomes the reverse inequality.

For a uniformly integrable martingale, boundedness of the stopping times can be removed, with the terminal limit used if a stopping time is infinite. A useful application is to a bounded stopped martingale $M_{t\wedge \tau }$ with $\tau <\infty$ almost surely: apply the bounded-time theorem first and then the bounded convergence theorem to obtain $\mathbb{E}{M}_{\tau }=\mathbb{E}{M}_0$. Finiteness of $\tau$ alone is not a sufficient hypothesis for an arbitrary martingale.

Let $X=(X^1,\dots ,X^d)$ be a continuous semimartingale, with $X^j=X_0^j+M^j+A^j$, where $M^j$ is a zero-starting continuous local martingale and $A^j$ is a continuous finite-variation process starting at zero. For $f\in C^{1,2}([0,\infty )\times {\mathbb{R}}^d)$, the Itô formula isHere $[M^j,M^k]$ is the quadratic covariation; the finite-variation process parts contribute no such term. The integrals against the finite-variation processes are understood pathwise, and those against the continuous local martingales are Itô integrals after localization.

For $d$-dimensional Brownian motion with independent standard coordinates, $d[B^j,B^k{]}_s={\delta }_{jk}ds$. In particular, if $f\in C^2$ is independent of time,where $\Delta$ is the Laplace operator. The same formulas hold up to exit from an open domain on which $f$ is smooth.

Let $\rho =|z_0|$ and $f(x,y)=\log \sqrt{x^2+y^2}$. On the annulus $r<\sqrt{x^2+y^2}<R$, its gradient and Laplace operator areThus $f$ is a harmonic function there. The stopped Itô formula givesThe sum of the squared coefficients is $|B_s{|}^{-2}\le r^{-2}$ before $\tau$. The Itô isometry therefore makes the stochastic term a true square-integrable martingale on every finite interval. Continuity prevents overshoot at exit, so $\log r\le \log |B_{t\wedge \tau }|\le \log R$. This proves the required bounded martingale property.

Before taking the terminal expectation, prove that $\tau$ is finite. Set ${\theta }_R=\inf \{t\ge 0:|B_t|=R\}$. The Itô formula for $|B{|}^2$ shows thatis a martingale, since its stopped stochastic integrands are bounded. Taking expectations yieldsThe monotone convergence theorem gives $\mathbb{E}{\theta }_R\le (R^2-{\rho }^2)/2<\infty$. Hence $\tau \le {\theta }_R<\infty$ almost surely.

The bounded convergence theorem, equivalently the optional stopping theorem applied to this bounded martingale, gives $\mathbb{E}\log |B_{\tau }|=\log \rho$. The two exit events partition the sample space up to a null set. Writing $p=\mathbb{P}({\tau }_r<{\tau }_R)$,Solving gives the planar Brownian annulus hitting probability

Fix an outer radius $R>\rho =|z_0|$. By continuity, reaching zero before radius $R$ requires first reaching every radius $\epsilon \in (0,\rho )$. The planar Brownian annulus hitting probability from (i) therefore impliesThus $\mathbb{P}({\tau }_0<{\tau }_R)=0$ for every fixed $R>\rho$.

On any continuous sample path that reaches zero in finite time, the radius has a finite maximum before that time. There is therefore some integer $m>\rho$ for which zero is reached before radius $m$. ConsequentlyEvery event on the right has probability zero, soThis proves that planar Brownian motion avoids a fixed point even though it can reach arbitrarily small positive radii.

First suppose $0<r<\rho =|z_0|$. For every $R>\rho$, the planar Brownian annulus hitting probability givesHence ${\tau }_r<\infty$ almost surely. This uses increasing outer radii; letting the inner radius tend to zero instead proves the different claim in part (ii).

For $r>\rho$, the disk-exit estimate in (i), with outer radius $r$, directly yields $\mathbb{E}{\tau }_r\le (r^2-{\rho }^2)/2<\infty$.

For $r=\rho$, retain the definition ${\tau }_r=\inf \{t>0:|B_t|=r\}$. The first case shows that radius $\rho /2$ is reached in finite time. Starting there, the Strong Markov property of Brownian motion and the disk-exit argument show that radius $\rho$ is subsequently reached in finite time. Thus the infimum over positive hitting times is finite in this case as well. Altogether,The conclusion can also be made simultaneous in all $r>0$: intersect the probability-one events for positive rational radii, then use continuity to cross each intermediate radius.

Let $f(y)=|y|$ away from zero. Its first and second partial derivatives areThus its Laplace operator is $\Delta f(y)=(n-1)/|y|$. The quadratic covariations of the independent Brownian coordinates satisfy $d[B_j,B_k{]}_s={\delta }_{jk}ds$. The stopped Itô formula therefore givesFor $t<\tau$, this is the required integral representation.

To identify the driving Brownian motion, the stopped stochastic term alone is insufficient: its quadratic variation is $t\wedge \tau$. Extend its unit direction after exit. Define a vector predictable process $u(s)$ byPredictability follows from the continuous adapted direction before stopping and the left-continuous adapted indicator $1_{\{s\le \tau \}}$. This $W$ is a zero-starting continuous martingale, andThe Lévy characterization of Brownian motion identifies $W$ as a one-dimensional Brownian motion. Before $\tau$, the preceding representation becomes the stochastic differential equation of the dimension-$n$ Bessel process:The fully stopped equation has both integrals stopped at $\tau$.

For $n=2$, apply the one-dimensional Itô formula to the logarithm. Before $\tau$, $d[X{]}_t=dt$, so the two drift terms cancel:Consequently,Indeed $X_s\ge r$ before stopping, so the Itô isometry bounds the stochastic term's second moment by $t/r^2$, and $\log X_{t\wedge \tau }\in [\log r,\log R]$. This is the martingale assertion under the paper's standing stopping assumption. Without stopping, the same calculation establishes only a local martingale; the stopped qualification is essential for the stated argument.