Past exam of the mathematics course of the University of Cambridge / 2007 / iii / Paper 36 / 2

The previsible sigma-algebra $\mathcal{P}$ on $\Omega \times [0,\infty )$ is generated by left-continuous adapted processes. Equivalently it is generated by $A\times \{0\}$ with $A\in {\mathcal{F}}_0$ and $A\times (s,t]$ with $A\in {\mathcal{F}}_s$, $0\le s<t$. A previsible process is a process measurable for this sigma-algebra; predictable is a synonym.

Use the zero-starting convention for ${\mathcal{M}}_c^2$: continuous martingales with $M_0=0$ and ${\sup }_t\mathbb{E}{M}_t^2<\infty$. For such an L2-bounded continuous martingale, the square-integrable stochastic integrand space isIdentify processes that agree almost everywhere for the quadratic-variation measure ${\nu }_M(d\omega ,ds)=\mathbb{P}(d\omega )d[M{]}_s$. This makes the displayed norm an actual Hilbert-space norm on equivalence classes. On a fixed horizon $[0,T]$, replace infinity by $T$; this is the same construction for continuous martingales square-integrable on that horizon. Nonzero initial values may be subtracted because the integral depends only on increments.

For an elementary previsible process $H={\sum }_{j=0}^{r-1}{H}_j{1}_{(t_j,t_{j+1}]}$, with bounded ${\mathcal{F}}_{t_j}$-measurable coefficients, defineThe Itô isometry isMore generally the squared $L^2$ distance between the integrals of $H$ and $K$ is the integral of $(H-K{)}^2$ on the right. The integral is a zero-starting continuous square-integrable martingale.

Elementary predictable processes are dense in $L^2(M)$. Given $H$ in this space, choose elementary $H_n$ converging in its norm and define the Itô integral by the $L^2$ limit of $H_n\cdot M$. The isometry proves that this is independent of the approximating sequence and remains valid for the limit. The Doob L2 maximal inequality bounds the expected squared path supremum of the difference by four times its terminal second moment; hence the convergence also gives a continuous martingale version, with uniform convergence on compacts in probability.

For a continuous local martingale $M$ and locally bounded process $H$ that is previsible, choose increasing stopping times ${\tau }_n\uparrow \infty$ such that $M^{{\tau }_n}$ is L2-bounded, $|H|\le K_n$ up to ${\tau }_n$, and ${\tau }_n\le n$. This can be done by combining localizers with first exits of $M$ from bounded intervals and the local bounds of $H$. Then $H{1}_{[0,{\tau }_n]}$ is in the integrand space for $M^{{\tau }_n}$. Define the integral on each stopped interval by the preceding construction. The stopping property of elementary integrals, extended by the isometry, makes these definitions agree on overlaps. Pasting givesThis is a localization construction; global square integrability of the resulting process is not asserted.

For every zero-starting continuous local martingale $M$, there is a unique, up to indistinguishability, continuous adapted increasing process $[M]$, with $[M{]}_0=0$, such thatThis is its quadratic variation. It also satisfies, for deterministic partitions whose mesh tends to zero,with convergence uniformly on compact time intervals in probability. The theorem includes existence and finiteness on each compact interval; monotonicity is understood pathwise outside one null set.

Uniqueness follows because the difference of two candidate increasing continuous processes is both a finite-variation process and a continuous local martingale, hence is identically zero from its zero initial value. For a continuous semimartingale $X=X_0+M+A$, its quadratic variation is $[X]=[M]$, since continuous finite-variation terms have zero quadratic variation. Polarization defines the quadratic covariation by $[M,N]=([M+N]-[M-N])/4$.

The correct quadratic variation of a stochastic integral identity isThe brackets around the final integrator are missing in the PDF. Without them the assertion is false: for $H=1$ and $M$ Brownian motion, the left side is $t$ while the printed right side is $B_t$.

For elementary predictable $H$, the integral is obtained by scaling each martingale increment by its known coefficient. Quadratic variation on each interval is therefore scaled by the square of that coefficient, giving the displayed formula. To pass to a general $L^2(M)$ integrand on $[0,T]$, take elementary $H_n\to H$ in the integrand norm and put $N_n=H_n\cdot M$, $N=H\cdot M$. The Itô isometry gives $\mathbb{E}(N_n-N{)}_T^2\to 0$, hence $\mathbb{E}[N_n-N{]}_T\to 0$. The bracket Cauchy-Schwarz boundand polarization show that $[N_n]\to [N]$ uniformly in probability. On the other side, Cauchy-Schwarz for the quadratic-variation measure givesTaking limits proves the formula in the square-integrable case. Stopping and pasting as in part b proves it for locally bounded predictable integrands and continuous local martingales.

For an elementary previsible integrand, the integral is the increment sum ${\sum }_j{H}_j(X_{t\wedge t_{j+1}}-X_{t\wedge t_j})$. This expression uses only $H$ and $X$, so is identical under the two measures, irrespective of how the local martingale and finite-variation parts change.

We may use the stochastic dominated convergence theorem: if bounded predictable $H_n$ converge pointwise to $H$ under a common deterministic bound, their integrals against a semimartingale converge uniformly on compact intervals in probability. Also convergence in $\mathbb{P}$-probability implies convergence in $\mathbb{Q}$-probability when $\mathbb{Q}\ll \mathbb{P}$. To see the latter directly, let $D=d\mathbb{Q}/d\mathbb{P}$. For any event $E$,and let first $\mathbb{P}(E)\to 0$, then $K\to \infty$. Apply this to events involving the supremum on a finite horizon for the uniform-in-probability statement.

Let $\mathcal{C}$ be the bounded predictable integrands for which the two integral processes agree up to $\mathbb{Q}$-indistinguishability. It is a vector space containing elementary rectangle indicators generating the predictable sigma-algebra. It is closed under uniformly bounded pointwise limits: stochastic dominated convergence theorem gives convergence of the integrals under each measure, and the preceding transfer makes the $\mathbb{P}$ limit also a $\mathbb{Q}$ limit. Uniqueness of limits in probability makes the two limiting integrals equal. The functional monotone-class theorem therefore puts every bounded predictable integrand in $\mathcal{C}$.

Finally, for locally bounded $H$, stop on intervals where it has a deterministic bound. Such stopping times tend to infinity under $\mathbb{P}$ and hence also under $\mathbb{Q}$. Apply the bounded result to the stopped integrands and use the stopping property of the integral. This proves the stochastic integral under an absolutely continuous measure change identityThe decompositions $H\cdot M+H\cdot A$ and $H\cdot N+H\cdot B$ thus represent the same process under $\mathbb{Q}$, although their individual components need not agree. With only absolute continuity, equality is asserted outside a $\mathbb{Q}$-null set, not necessarily outside a $\mathbb{P}$-null set for a chosen $\mathbb{Q}$ version.