Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 202 (Stochastic Calculus and Applications)

The integrand $\mathrm{sign}({\beta }_s)$ is a bounded previsible process, so $B$ is a continuous local martingale. The quadratic variation of a stochastic integral isbecause the Brownian zero set has zero Lebesgue measure. Since $B_0=0$, the Lévy characterization of Brownian motion shows that $B$ is a standard Brownian motion.

Both variables are centered. The Itô isometry in its bilinear form giveswhere the last equality follows because a centered Gaussian distribution is a symmetric probability distribution. Thus $B_t$ and ${\beta }_t$ are uncorrelated random variables.

They are not independent. The Itô formula gives ${\beta }_t^2=t+2{\int }_0^t{\beta }_{s}d{\beta }_s$, and the bilinear Itô isometry therefore givesIf $B_t$ and ${\beta }_t$ were independent random variables, then $B_t$ would also be independent of the measurable function ${\beta }_t^2$, and centeredness would instead give $\mathbb{E}[B_t{\beta }_t^2]=\mathbb{E}[B_t]\mathbb{E}[{\beta }_t^2]=0$. This contradiction disproves independence.

The Dambis-Dubins-Schwarz theorem states that if $M$ is a continuous local martingale with $M_0=0$ and $[M{]}_{\infty }=\infty$, then, forthe process $W_s=M_{{\tau }_s}$ is a standard Brownian motion and $M_t=W_{[M{]}_t}$. If $[M{]}_{\infty }<\infty$, one obtains the same representation after enlarging the probability space and continuing $W$ independently beyond $[M{]}_{\infty }$.

Set $M_t={\int }_0^t{H}_{s}d{B}_s$. Its quadratic variation is $[M{]}_t={\int }_0^t{H}_s^{2}ds$, which is continuous and tends to infinity almost surely by assumption. The stated stopping time is the inverse clock at level one, so $[M{]}_{\tau }=1$. The Dambis-Dubins-Schwarz theorem gives

The assertion is false: the Brownian motion produced by the Dambis-Dubins-Schwarz theorem need not be independent of its clock. Let $W$ be a standard Brownian motion and setIf the Brownian motion $B$ in $M_t=B_{[M{]}_t}$ were independent of the whole quadratic variation process, then conditioning on $[M]$ would give $\mathbb{E}[M_t[M{]}_t]=\mathbb{E}[B_{[M{]}_t}[M{]}_t]=0$. Instead, the fourth-moment formula for a bivariate normal distribution gives $\mathrm{Cov}(W_t^2,W_s^2)=2s^2$ for $s\le t$, and henceThus $B$ and $[M]$ are dependent.

Because independent Brownian motions have zero quadratic covariation, the Itô product rule givesAfter integration, the random variable in the question is $B_t^1{B}_t^2$. Writing $B_t^j=\sqrt{t}{Z}_j$ for independent standard Gaussian random variables $Z_1,Z_2$, its distribution isthe scaled product of two independent standard normal random variables.

A càdlàg process $A$ is a finite-variation process when, almost surely, for every $T<\infty$,where the supremum is over every finite partition of an interval $0=t_0<\cdots <t_n=T$.

Uniform convergence on compacts in probability of $X^n$ to $X$ means that, for every $T<\infty$ and $\epsilon >0$,

Apply the realized absolute covariation theorem to each dyadic partition of an interval. More explicitly, use the continuous increasing clock $C=[M]+[N]$ and the Radon-Nikodym theorem to writeFor each time, let $(U,V)$ have the centered bivariate normal distribution with covariance matrix $\left(\begin{array}{cc}a& c\\ c& b\end{array}\right)$, and defineThis process is continuous and increasing. To prove convergence, localize $M,N$, represent the pair as stochastic integrals against a two-dimensional Brownian motion, and approximate the integrands in $L^2(dC)$ by bounded step previsible processes. For step integrands, the result is the weak law of large numbers applied on each block to independent Gaussian random variables. The Burkholder-Davis-Gundy inequality and the Cauchy-Schwarz inequality make the error uniform on each compact interval in probability. Consequentlyin the sense of uniform convergence on compacts in probability.

With the notation from part (i), the total-variation process of $[M,N]$ isFor the centered bivariate normal distribution $(U,V)$ used there, $c=\mathbb{E}[UV]$. The integral triangle inequality givesIntegrating this pointwise inequality against $dC$ proves $V_t\le {\tilde{V}}_t$ for every $t\ge 0$.

The Kunita-Watanabe inequality applied to the continuous local martingales $M,N$ gives directlyEquivalently, with the clock and densities from part (i), positivity of the covariance matrix gives $|c|\le \sqrt{ab}$, and the Cauchy-Schwarz inequality gives

The martingale product identity says that $N_t{K}_t-[N,K{]}_t$ is a martingale. Passing to the terminal values of the square-integrable martingales and using $N_0=K_0=0$ givesThe quadratic covariation identity for a stochastic integral isApplying the same product identity to $H\cdot M$ and $K$ therefore gives

Let $L=N-H\cdot M$. It is a continuous square-integrable martingale, and the assumed bracket identity givesfor every continuous square-integrable martingale $K$. Choose $K=L$ and use part (i):Thus $L_{\infty }=0$ almost surely, and the conditional expectation property gives $L_t=\mathbb{E}[L_{\infty }\mid {\mathcal{F}}_t]=0$ for every $t$. Hence $N=H\cdot M$ up to indistinguishability of stochastic processes.

Fix $0\le r\le t$. Since $M$ is a martingale,Every finite vector consisting of $M_{t+s}-M_t$ and past values $M_{r_1},\dots ,M_{r_n}$ has a multivariate normal distribution. Therefore uncorrelated jointly normal variables are independent, so the increment is independent of every finite vector of past values. A Monotone class theorem then extends this to independence from ${\mathcal{F}}_t=\sigma (M_r:0\le r\le t)$. This is the independent increments of a Gaussian martingale.

Define the deterministic function $f(t)=\mathbb{E}[M_t^2]$. The independent increments from part (i) show that $f$ is increasing and that $M_t^2-f(t)$ is a martingale. Mean-square continuity follows from path continuity and the Gaussian laws, so $f$ is continuous.

The Itô formula also says that $M_t^2-[M{]}_t$ is a local martingale. Their difference $[M{]}_t-f(t)$ is therefore a continuous finite-variation process that is also a local martingale. By the theorem that a continuous finite-variation local martingale is constant, and because the difference starts at zero,for all $t\ge 0$ almost surely.

The assertion is false. Let $Z\sim N(0,1)$ and define $X_t=tZ$. This is a centered continuous Gaussian process. Its natural filtration satisfies $Z=X_s/s\in {\mathcal{F}}_s^X$ for every $s>0$, and hence, for $0<s<t$,with positive probability. Thus $X$ is not a martingale and does not belong to the stated martingale class.

The assertion is true. The Dambis-Dubins-Schwarz theorem, with an independent continuation of the Brownian motion if $f$ is bounded, representsBecause $f$ is deterministic, every finite vector $(M_{t_1},\dots ,M_{t_n})$ is a finite vector of a Brownian motion at deterministic times and therefore has a multivariate normal distribution. Hence $M$ is a Gaussian process. This is the deterministic quadratic variation characterizes a Gaussian continuous local martingale result.

The function $z\mapsto \log |z|$ is a harmonic function on the annulus $r_1<|z|<r_2$. The Itô formula therefore makes $\log |B_{t\wedge \tau }|$ a bounded martingale. The optional sampling theorem for a supermartingale giveswhere $p=\mathbb{P}(|B_{\tau }|=r_1)$. Solving this linear equation yields the planar Brownian annulus hitting probability

No such function exists. Continuity on the compact closed unit disk makes $u$ bounded near the origin. The removable singularity for a bounded harmonic function extends $u$ harmonically across the origin. The extended function is continuous on the closed disk and vanishes on its boundary, so the maximum principle for harmonic functions, applied to both $u$ and $-u$, forces $u=0$ throughout the disk. This contradicts $u(0,0)=1$.

On $[\tau ,\sigma )$ the Brownian path does not meet zero, so the power function $x\mapsto |x{|}^{\delta }$ is twice continuously differentiable along the path. The Itô formula givesSince $Y_t=|B_t{|}^{\delta }$, this becomeswhere ${\hat{B}}_t={\int }_0^t\mathrm{sign}(B_s)d{B}_s$ is a standard Brownian motion by the Lévy characterization of Brownian motion. Thus the displayed equation in the paper is valid after the customary renaming of $\hat{B}$ as $B$.

LetThe clock $A$ is an absolutely continuous function and is strictly increasing: its derivative is positive away from the Brownian zero set, which has zero Lebesgue measure. It also tends to infinity. This is immediate for $\delta =1$; for $\delta >1$, recurrence and the Strong Markov property imply that the Brownian occupation time of, for example, $\{x:1<|x|<2\}$ is unbounded, while the integrand is bounded below there by ${\delta }^2$.

Thus $A$ is continuous, strictly increasing, and maps $[0,\infty )$ onto itself. Its inverse function ${\tau }_s=A^{-1}(s)$ is finite, continuous, and strictly increasing.

Part (b) shows that $s\mapsto {\tau }_s$ is finite and continuous. Since both $t\mapsto B_t$ and the power function $x\mapsto |x{|}^{\delta }$ are continuous, their compositionis continuous. This is an instance of a time change of a continuous process.

As printed, the requested conclusion is false for $\delta \ne 1$. On an interval on which $X$ stays positive, the time change of a continuous process satisfies $d{\tau }_s=ds/({\delta }^2{X}_s^{2(\delta -1)/\delta })$. The time-changed martingale term has quadratic variation $s$, so the Lévy characterization of Brownian motion identifies it with a standard Brownian motion $\tilde{B}$. Dividing the drift in part (a) by the derivative of the clock givesConsequently the construction actually satisfieswhich is the Bessel process equation of dimension $2-1/\delta$. It equals the paper's claimed drift $(\delta -1)/(2X_s)$ only when $\delta =1$. The mismatch between the specified power, clock, and conclusion is therefore a typographical error in the question.

For the equation actually produced by the preceding construction, namely the Bessel process equation of dimension $2-1/\delta$, the drift has Lipschitz continuity on every compact subset of $(0,\infty )$. Starting at any positive time and position, pathwise uniqueness therefore makes the time-changed process agree until its first hit of zero with the maximal local solution of a stochastic differential equation. For $\delta >1$, its dimension lies in $(1,2)$, so it can hit zero; the time-change construction then supplies further excursions, whereas the maximal local solution on $(0,\infty )$ stops at that first hit.

For $\delta =1$, $X_s=|B_s|$ is Reflected Brownian motion; away from zero it agrees with the maximal local solution of $dX=d\tilde{B}$. For the dimension-$\delta$ equation printed in the paper, the preceding construction does not agree with the maximal local solution unless $\delta =1$, for the coefficient mismatch established in part (d).

Let $Z=\{t:B_t=0\}$. Since $A$ is the clock from part (b) and $\tau =A^{-1}$,For $\delta >1$, the absolutely continuous function $A$ has derivative $A^{\prime }(t)={\delta }^2|B_t{|}^{2\delta -2}=0$ on $Z$. The one-dimensional area bound for an absolutely continuous function therefore givesFor $\delta =1$, $A_t=t$ and the conclusion follows directly because the Brownian zero set has zero Lebesgue measure. Hence the zero set of $X$ has zero Lebesgue measure almost surely.

The relation $P\ll \tilde{P}$ is an instance of absolute continuity of measures: it means that every $\tilde{P}$-null event is also $P$-null,By the Radon-Nikodym theorem, this is equivalent to the existence of a nonnegative Radon-Nikodym derivative $dP/d\tilde{P}$ whose $\tilde{P}$-expectation is one.

For a continuous local martingale $M$ with $M_0=0$, its stochastic exponential isIt is the unique solution of the stochastic differential equation $d{Z}_t=Z_{t}d{M}_t$, $Z_0=1$, and is a nonnegative local martingale.

One continuous form of the Girsanov theorem is as follows. Let $M$ be a continuous local martingale under $P$, and suppose $Z=\mathcal{E}(M)$ is a true martingale on $[0,T]$. Define $Q$ by $dQ=Z_{T}dP$. Then every continuous $P$-local martingale $N$ becomes the continuous $Q$-local martingaleIn particular, if $M_t={\int }_0^t{\theta }_{s}d{W}_s$, then ${\tilde{W}}_t=W_t-{\int }_0^t{\theta }_{s}ds$ is a $Q$-Brownian motion.

Work first under Wiener measure $P$ with coordinate Brownian motion $X$. Boundedness of $b$ implies the Novikov condition, sohas expectation one. Define $Q$ by $dQ=Z_{T}dP$. The Girsanov theorem makesa $Q$-Brownian motion, and hence $(X,W,Q)$ is a weak solution of a stochastic differential equation.

For uniqueness in law, start with any weak solution under $Q$ and apply the inverse change of measure with density $\mathcal{E}(-\int b(X_s)d{W}_s{)}_T$. Boundedness again gives the Novikov condition, and under the resulting measure $P$ the process $X$ is Brownian. Reversing the density expresses the law of $X$ under $Q$ as the same functional $Z_T$ of a Wiener path. It is therefore independent of the chosen weak solution. This proves the Weak existence and uniqueness in law for an additive-noise SDE with bounded drift.

The assertion is false. For a common sequence, each term of which is a refining deterministic partition of an interval, standard Brownian paths have quadratic variation $t$ almost surely, whereas the paths $t\mapsto B_{2t}$ have quadratic variation $2t$ almost surely. These two path properties define disjoint measurable subsets of $C([0,1])$, so the two laws are mutually singular measures. In particular, the law of $B_{2\cdot }$ is not absolutely continuous with respect to Wiener measure. This is the Pathwise quadratic variation distinguishes Brownian speeds argument.

The assertion is true. Since $f\in C^1[0,1]$, it lies in the Cameron-Martin space of Wiener measure. The Cameron-Martin theorem says that the translated law is equivalent, and in particular absolutely continuous, with respect to Wiener measure. Its Radon-Nikodym derivative is

The assertion is false for a general continuous $f$. For example, take $f(t)=t^{1/3}$. The Brownian Hölder regularity gives $B_t/t^{1/3}\to 0$ almost surely as $t\downarrow 0$, whilealmost surely. The original and translated path laws therefore concentrate on disjoint measurable events and are mutually singular measures.

More generally, the Cameron-Martin theorem gives the exact criterion: translation by $f$ is absolutely continuous precisely when $f$ is an absolutely continuous function, $f(0)=0$, and $f^{\prime }\in L^2[0,1]$.

Fix $t>0$ and define $w(s,y)=u(t-s,|y|)$ for $0\le s\le t$. The assumed smooth extension and the Neumann boundary condition at zero make $w$ a $C^{1,2}$ function. The heat equation givesThe Itô formula therefore makes $w(s,B_s)$ a local martingale. Stop first when $|B|$ leaves a large compact interval. The exponential growth bound and the finite exponential moments of the maximum of Brownian motion on $[0,t]$ give uniform integrability, so localization and the dominated convergence theorem yieldThis is the Feynman-Kac formula for the Neumann heat problem.

Let $p_t(z)=(2\pi t{)}^{-1/2}{e}^{-z^2/(2t)}$ be the heat kernel for $u_t=\frac{1}{2}{u}_{xx}$. The symmetry of the Gaussian distribution gives the method-of-images formulaThis is the Neumann heat kernel on a half-line. Differentiation under the integral sign shows that $v\in C^{1,2}$ for $t,x>0$ and that $v_t=\frac{1}{2}{v}_{xx}$. At $x=0$, the two differentiated kernel terms cancel, so $v_x(t,0)=0$. The Gaussian approximate identity gives $v(t,x)\to f(x)$ as $t\downarrow 0$, while the dominated convergence theorem gives continuity up to $x=0$. Finally $|v(t,x)|\le \Vert f{\Vert }_{\infty }$, which is stronger than the required exponential bound. Thus $v$ satisfies every condition in the displayed boundary-value problem.

Let $\rho =t\wedge {\tau }_0\wedge {\tau }_1$ and apply the Itô formula to $u(t-s,B_s)$ for $0\le s\le \rho$. The heat equation cancels the drift, so the stopped process is a bounded martingale. The optional sampling theorem for a supermartingale gives $u(t,x)={\mathbb{E}}_x[u(t-\rho ,B_{\rho })]$. On the three mutually exclusive terminal events, the initial and Dirichlet boundary conditions identify this value asThis is the probabilistic representation of the heat equation with time-dependent Dirichlet data.