Past exam of the mathematics course of the University of Cambridge / 2016 / iii / Paper 202 / 1

Use a filtered probability space $(\Omega ,\mathcal{F},({\mathcal{F}}_t),\mathbb{P})$, and put ${\mathcal{F}}_{0-}={\mathcal{F}}_0$. The predictable sigma-algebra on $\Omega \times [0,\infty )$ isEquivalently it is generated by left-continuous adapted processes. A real-valued previsible process is a function $(\omega ,t)\mapsto H_t(\omega )$ that is measurable for $\mathcal{P}$ and the Borel sigma-algebra of $\mathbb{R}$.

A simple predictable process has the formwhere $h_{-1}$ is bounded and ${\mathcal{F}}_0$-measurable, and each $h_i$ is bounded and ${\mathcal{F}}_{t_i}$-measurable. Set $H=0$ after $t_n$; allowing a final infinite interval gives an equivalent dense class. The values may be restricted to finitely many values if desired, by approximating bounded measurable functions by simple functions. The initial-time term matters for arbitrary finite measures on $\mathcal{P}$: a measure can charge time zero, even though the stochastic integral against a continuous martingale does not.

For fixed $u>0$, define the left-limit sigma-algebra by ${\mathcal{F}}_{u-}=\sigma ({\bigcup }_{s<u}{\mathcal{F}}_s)$. The section at $u$ of a generating rectangle $A\times (s,t]$ of the predictable sigma-algebra is $A$ when $s<u\le t$, and empty otherwise. In the first case $A\in {\mathcal{F}}_s\subseteq {\mathcal{F}}_{u-}$. Sections of the time-zero generators are empty for $u>0$.

The collection of subsets of sample-time space whose section at $u$ belongs to ${\mathcal{F}}_{u-}$ is a sigma-algebra: sections commute with complements and countable unions. It therefore contains all of $\mathcal{P}$. Apply this to $\{(\omega ,t):H_t(\omega )\in E\}$ for every Borel set $E$. Thus predictable sections at deterministic times satisfyAt $u=0$ the same argument gives ${\mathcal{F}}_0$-measurability, using the convention ${\mathcal{F}}_{0-}={\mathcal{F}}_0$. The PDF asks for information strictly before $u$; the TeX aid drops the minus sign.

Let $V$ be the closure of the simple predictable processes in the Hilbert space $L^2(\mathcal{P},\mu )$. Common refinement of deterministic partitions makes these processes a vector space. To prove density of simple predictable processes for finite measures, suppose $g\in V^{\perp }$. Since $\mu$ is a finite measure, the Cauchy-Schwarz inequality gives $g\in L^1(\mu )$, sodefines a finite signed measure. The indicator of each predictable generating rectangle, including each time-zero rectangle, is a simple predictable process; hence $\nu$ vanishes on these rectangles.

Finite intersections of the rectangles are again rectangles or empty. Add the whole space to this pi-system. Its $\nu$-mass is zero because $1_{\Omega \times [0,n]}$ is simple and converges to $1$ in $L^2(\mu )$ by the dominated convergence theorem. The zero sets of $\nu$ form a lambda-system now that its total mass is zero. The Pi-lambda theorem therefore gives $\nu (E)=0$ for every $E\in \mathcal{P}$. Taking the sets where $g>0$ and $g<0$ gives $g=0$ $\mu$-almost everywhere.

Thus $V^{\perp }=\{0\}$ and the orthogonal complement criterion for a closed subspace of a Hilbert space yieldsOmitting the time-zero term would make this conclusion false when $\mu$ is concentrated at time zero.

For the simple predictable process from part (a), define its stochastic integral byThe time-zero value contributes nothing. Write $D_i=M_{t\wedge t_{i+1}}-M_{t\wedge t_i}$. If $i<j$ and $t>t_j$, then $h_i{D}_i{h}_j$ is ${\mathcal{F}}_{t_j}$-measurable and $\mathbb{E}[D_j\mid {\mathcal{F}}_{t_j}]=0$. If $t\le t_j$, then $D_j=0$. Boundedness ensures integrability, so all off-diagonal terms vanish by conditional expectation. ConsequentlyThe quadratic variation theorem says $M^2-[M]$ is a local martingale. A bounded continuous martingale is square-integrable, and localization gives the conditional second-moment identityMultiplication by $h_i^2$ and summation therefore give the Itô isometryFor Brownian motion, $[M{]}_s=s$. More generally the Itô isometry holds for every predictable process with finite right-hand side, against a continuous square-integrable martingale. Indeed, on a fixed horizon the quadratic-variation measure ${\mu }_M(E)=\mathbb{E}{\int }_0^t{1}_{E}d[M]$ is finite. Part (c) approximates the integrand by simple predictable processes; the displayed equality makes the elementary integrals Cauchy in $L^2$ and defines their unique limit. The finite sum is exactly the elementary case of the Itô isometry.