Past exam of the mathematics course of the University of Cambridge / 2010 / iii / Paper 29 / 1

Set $h_n=2^{-n}$ and $t_n=h_n\lceil t/h_n\rceil$. Replace the last sampled value $X_{t_n}$ by $X_t$, and call the resulting absolute-increment sum $W_n(t)$. This is a sum over a genuine partition of an interval of $[0,t]$, andby right continuity. This is the dyadic approximation of total variation. Let $V(t)$ be the total variation of a function $X$ on $[0,t]$, allowing $+\infty$. Every $W_n(t)$ is at most $V(t)$. Conversely, take any finite partition of an interval of $[0,t]$. Approximate its interior points from the right by dyadic grid points, retaining the endpoint $t$. For large $n$ these approximating points remain distinct, and right continuity makes their absolute-increment sum converge to that of the original partition. The triangle inequality makes $W_n(t)$ at least this selected sum. Taking the supremum over all partitions provesThe limit is the total-variation function; for locally finite variation it is the total-variation process.

There is a missing hypothesis in the assertion about being càdlàg. A general càdlàg function can have infinite total variation of a function arbitrarily close to zero. For example,is continuous. At the successive points $s_j=(\pi /2+j\pi {)}^{-1}$ its values alternate in sign with magnitude $s_j$. The variation along these points dominates a divergent harmonic series. Thus $V(t)=\infty$ for every $t>0$, but $V(0)=0$, so even extended-valued right continuity fails at zero.

The correct càdlàg conclusion holds when $X$ has finite variation on every compact interval. Indeed $V$ is then finite and nondecreasing, so it has finite left limits. To prove right continuity at $t$, fix $s>t$ and choose a partition of $[t,s]$ whose sum is within $\epsilon$ of its variation. If $u>t$ lies before the first interior partition point, changing the first endpoint from $t$ to $u$ changes that sum by at most $|X_u-X_t|$. Additivity of total variation of a function therefore givesLet $u\downarrow t$ and then $\epsilon \downarrow 0$. This proves right continuity. This proves càdlàg regularity of finite total variation. Under that qualification, the jump formula isIf the variation is infinite, a difference of two infinite variation values is not a defined jump.

Fix $T<\infty$. By uniform continuity of each path of the continuous local martingale on $[0,T+1]$, its modulustends to zero almost surely. For every $s\le T$, comparison of powers of each increment givesThe squared-increment sum converges in probability to $[X{]}_T$, by the dyadic definition of quadratic variation. Including the last full grid interval instead of the last partial interval changes that sum by a quantity bounded by $2{\eta }_n^2$, so the ceiling convention has the same limit. In particular these squared-increment sums are bounded in probability.

Since $p-2>0$, a quantity tending to zero almost surely multiplied by a family bounded in probability tends to zero in probability. HenceThis is uniform convergence on compacts in probability, and proves the superquadratic case of dyadic power variation of a continuous local martingale.

Fix the specified time $t$. With the path modulus ${\eta }_n$ on $[0,t+1]$ as above, comparison of increment powers now givesThe hypothesis makes the second factor eventually bounded almost surely, while the first tends to zero almost surely because $2-p>0$. Thus the squared-increment sums tend to zero almost surely. They also converge in probability to the quadratic variation $[X{]}_t$, so uniqueness of the limit in probability implies $[X{]}_t=0$ almost surely.

The quadratic variation is nondecreasing, hence it vanishes throughout $[0,t]$. To see explicitly that the continuous local martingale must then be constant, localize it to square-integrable stopped martingales. The Itô isometry givesRemove the localization, apply the conclusion at every rational $s\le t$, and use continuity. Since $X_0=0$, this provesThus $X$ is indistinguishable from the zero process on that interval. This is the subquadratic case of dyadic power variation of a continuous local martingale.

No quadratic variation argument is needed. For deterministic $u<v$, the martingale property and square integrability giveConsequentlyApply this identity separately to each consecutive grid increment. The second moments telescope, givingThis uniform bound follows directly from martingale-difference orthogonality, including the final interval specified by the ceiling.