Past exam of the mathematics course of the University of Cambridge / 2012 / iii / Paper 34 / 2

For a continuous path $m$ of finite variation on $[0,t]$, every partition satisfiesUniform continuity makes the right side tend to zero as the mesh tends to zero. The quadratic variation of the given continuous local martingale $M$ is therefore identically zero.

Stop $M$ when its magnitude first reaches an integer $n$, so the stopped process is a bounded, square-integrable martingale. The square identity, or the Itô isometry, gives $\mathbb{E}{M}_{t\wedge {\tau }_n}^2=\mathbb{E}[M{]}_{t\wedge {\tau }_n}=0$. Hence it is zero at each rational time almost surely, and continuity extends this simultaneously to all times. Let $n$ tend to infinity. This provesThis is the continuous finite-variation local martingale is constant principle with initial value zero.

For a continuous semimartingale $X$, its quadratic variation is the continuous increasing zero-starting process obtained as the limit, uniformly on compact time intervals in probability, of sums of squared increments along deterministic partitions whose mesh tends to zero. If $X=X_0+N+V$ is its continuous local-martingale/finite-variation decomposition, then $[X]=[N]$; finite-variation terms and their cross sums vanish.

Write $D=d\mathbb{Q}/d\mathbb{P}$. If $\mathbb{P}(E_n)\to 0$, then for every $R>0$,First send $n$ to infinity and then $R$ to infinity. Thus convergence in probability under $\mathbb{P}$ implies convergence in probability under $\mathbb{Q}$, also for a supremum on a compact time interval. The same squared-increment sums converge under $\mathbb{Q}$ to the $\mathbb{P}$ version of $[X]$. Since $X$ is also a semimartingale under $\mathbb{Q}$, these sums converge to its $\mathbb{Q}$ quadratic variation. Uniqueness of limits in probability gives equality at every fixed time; continuity and countably many rational times giveAbsolute continuity suffices; equivalence is not required. This is quadratic variation under an absolutely continuous measure change.

Use just the parameters $1$ and $-1$. Positivity and adaptation of their local martingales giveThus both processes are adapted. A positive continuous local martingale has a semimartingale logarithm, by Itô formula after localization away from zero. Since $X=\log Z^{(1)}+A/2$ and $A$ has finite variation, $X$ is a continuous semimartingale. Write $X=N+V$, where $N$ is a zero-starting continuous local martingale and $V$ is continuous finite variation, also starting from zero.

Apply Itô formula to each exponential:Because $Z^{(\theta )}$ is itself a local martingale, the finite-variation term is zero by part (a). Divide its finite signed measure by the strictly positive $Z^{(\theta )}$. For $\theta =1,-1$, respectively,Adding and subtracting show $V=0$ and $[N]=A$. Therefore the exponential criterion for a continuous local martingale and its bracket givesThe semimartingale property was established before applying Itô's formula to $X$; it was not assumed from the outset.