Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 202 (Stochastic Calculus and Applications)

The L2 martingale convergence theorem gives $M_t\to M_{\infty }$ in $L^2$, soConversely, apply the Doob L2 maximal inequality on $[0,T]$:Monotone convergence as $T\to \infty$ givesThus the two norms are equivalent.

For a simple predictable processdefineEach summand is a bounded predictable multiple of a martingale increment, so conditional expectation proves that $H\cdot M$ is a martingale. Orthogonality of disjoint martingale increments givesIt is therefore an $L^2$-bounded continuous martingale.

Indicators of the rectanglestogether with $A\times \{0\}$ generate the predictable sigma-algebra. Their finite linear span is precisely the set of simple processes. The monotone-class theorem therefore makes this span dense among bounded predictable functions in measure. Since $\nu$ is finite, truncation followed by bounded approximation proves density in $L^2(\mathcal{P},\nu )$.

Let ${\mathcal{M}}_c^2$ be the continuous $L^2$-bounded martingales starting at zero, modulo indistinguishability, with norm $\Vert N{\Vert }_{{\mathcal{M}}^2}=\Vert N_{\infty }{\Vert }_2$. For a fixed $M\in {\mathcal{M}}_c^2$, define a finite measure on $\mathcal{P}$ byand let $L^2(M)=L^2(\mathcal{P},{\nu }_M)$. The Itô isometry is the isometric extensionsatisfying

For the simple process in part b, orthogonality gives the sum there. Conditional on ${\mathcal{F}}_{t_i}$, the martingale identity for $M^2-[M]$ givesSumming proves the isometry. Part c then supplies the unique extension to all of $L^2(M)$.

A process is previsible when it is measurable with respect to the predictable sigma-algebra. The deterministic process $H_t=1_{[1,\infty )}(t)$ is predictable because deterministic Borel processes are predictable, but it is not left-continuous at $t=1$ on any sample path.

Since $|Z_t{Y}_t|\le C{Z}_t$ and $Z$ is uniformly integrable, the continuous local martingale $ZY$ is locally in Doob's class and hence is a true martingale. For $A\in {\mathcal{F}}_s$ and $s\le t$, Bayes formula for conditional expectation givesThe bounded process $Y$ is integrable under $\tilde{\mathbb{P}}$, so this is exactly the martingale property.

The Cameron-Martin theorem on Wiener space says that the translated measure ${\mathbb{P}}_h(A)=\mathbb{P}(X+h\in A)$ is equivalent to Wiener measure precisely when $h$ is absolutely continuous, $h(0)=0$, and $\dot{h}\in L^2({\mathbb{R}}_{+})$. In that case

For such $h$, the exponential is a uniformly integrable stochastic exponential. Under the measure defined by this density, the Girsanov theorem makes $X_t-{\int }_0^t{\dot{h}}_{s}ds=X_t-h(t)$ a Brownian motion. This identifies the translated law and proves equivalence; replacing $h$ by $-h$ gives the inverse density.

If $h$ fails the Cameron-Martin condition on some finite interval, the finite-horizon theorem gives singularity there. If it belongs locally but ${\int }_0^{\infty }{\dot{h}}_s^{2}ds=\infty$, the log likelihood is a Brownian motion run at that diverging energy clock minus half the clock. It tends to $-\infty$ under one measure and to $+\infty$ under the translate, producing disjoint full-measure events. Thus the measures are singular.

The Cameron-Martin theorem for a linear drift changes the density of Brownian paths through time $t$ byAt the driftless hitting time ${\tau }_{a,0}=t$, the endpoint is $B_t=a$. Multiplying its given density by the likelihood $e^{ab-b^{2}t/2}$ therefore yields

Let ${\tau }_n$ localize $M$. The Itô isometry givesThus the stopped variables are bounded in $L^2$, and localization plus weak compactness shows that $M$ is a true martingale. The Itô formula applied to $f(x)=x^2$ givesAfter localization this is a martingale; the Burkholder-Davis-Gundy inequalities and $\mathbb{E}[M{]}_t<\infty$ supply the required local integrability, so it is a true martingale.

Part 1 gives $\mathbb{E}{M}_t^2=\mathbb{E}[M{]}_t$. Henceso $M$ is $L^2$-bounded.

The martingale product identity and the Itô isometry for cross terms giveSince $B_s\sim N(0,s)$, its moment-generating function gives $\mathbb{E}{e}^{B_s}=e^{s/2}$. Therefore the answer is

Let $X^x$ solveThe Feynman-Kac formula isFix $t$ and apply the two-variable Itô formula to $F(s,y)=u(t-s,y)$ and the semimartingale vector $(s,X_s)$. Multiplying byand using the Itô product rule, the drift of $R_{s}F(s,X_s)$ isThe remaining stochastic integral is a true martingale because the coefficients and derivatives are bounded. Taking expectations at $s=0$ and $s=t$ gives the formula.

Applying the Itô formula to $f(x)=x^2$ and the semimartingale $B$ givesHenceThe first term is a continuous local martingale and the second has finite variation. By uniqueness of the continuous semimartingale decomposition, $X$ could be a local martingale only if the finite-variation term were constant. Its derivative $B_s^2-1$ is not zero almost everywhere, so $X$ is not a local martingale.

Enlarge the space by an independent Brownian motion $W$ and defineThe two terms have zero cross-variation andThe Lévy characterization of Brownian motion makes $B$ a Brownian motion. The residual $\int 1_{\{A=0\}}dX$ has zero quadratic variation and is therefore constant, so

An $L$-diffusion solves the martingale problem forfor every $f\in C_c^2(\mathbb{R})$,is a local martingale. Applying this to cutoff approximations of $x$ and $x^2$ shows thatis a continuous local martingale withPart b gives $M=\int |\sigma (X_s)|d{W}_s$. Changing the sign of $W$ predictably where $\sigma <0$, and filling the zero set with independent Brownian noise, produces a Brownian motion $B$ such that $M=\int \sigma (X_s)d{B}_s$. Thus

For every real $\lambda$, positivity of quadratic variation givesThe discriminant of this quadratic is nonpositive, soContinuity lets the almost-sure assertion hold simultaneously for every $t$.

For a partition $\pi$ of $[0,t]$, apply the first inequality to each increment of the covariation and then the Cauchy-Schwarz inequality:Taking the supremum over partitions proves the Kunita-Watanabe inequality

Let ${\tau }_n$ localize the nonnegative local martingale $M$. For $s\le t$,Conditional Fatou lemma and nonnegativity giveThus $M$ is a supermartingale, recovering the general fact about a nonnegative local martingale.

A strong solution of a stochastic differential equation is adapted to the completed filtration of a prescribed Brownian motion $B$ on a prescribed probability space and satisfiesalmost surely. A weak solution of a stochastic differential equation may choose the filtered probability space, Brownian motion, and adapted process as part of the solution; only the displayed integral equation and the prescribed initial law are required.

Apply the Itô formula to $f(t,x)=x_0{e}^{\sigma x+(\mu -{\sigma }^2/2)t}$ and the semimartingale vector $(t,B_t)$. Its derivatives giveThereforeThis is adapted to the given Brownian filtration and is consequently a strong solution; it is geometric Brownian motion.

Approximate $h$ in $L^2[0,t]$ by deterministic step functions $h_n$. Each integral $\int h_{n}dB$ is a linear combination of independent Gaussian increments and hence is Gaussian, with mean zero and variance $\int h_n^2$. The Itô isometry gives convergence in $L^2$ to $\int hdB$, so characteristic functions pass to the limit. Thus

The Itô product rule for $e^{\lambda t}{X}_t$ givesTherefore the Ornstein-Uhlenbeck process has the explicit form

Part a applied to the deterministic kernel gives

Every linear combination of $X_{t_1},\dots ,X_{t_n}$ is a deterministic constant plus one stochastic integral of a deterministic $L^2$ function against $B$. Part a makes every such combination Gaussian. By the linear-combination characterization of a multivariate normal distribution, $(X_{t_1},\dots ,X_{t_n})$ is jointly Gaussian, so $X$ is a Gaussian process.

For $0<s<t$, only the Brownian noise accumulated through time $s$ is shared. The Itô isometry for cross terms gives

If $X_0\sim N(0,(2\lambda {)}^{-1})$ independently of $B$, thenThus $X_t\sim N(0,(2\lambda {)}^{-1})$ for every $t$. For $0<s<t$, the Markov decompositionhas an increment independent of $X_s$, and henceThis is the stationary Ornstein-Uhlenbeck covariance.