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For a càdlàg function, replace the last sampled value at $2^{-n}\lceil 2^{n}t\rceil$ by $X_t$. Right continuity makes this change tend to zero. The corrected sum is bounded by the total variation of a function. Conversely every finite partition of an interval can be approximated by dyadic points from the right, retaining the endpoint $t$. The triangle inequality bounds that partition's limiting sum by the dyadic sums. Taking the supremum proves convergence, including the value $+\infty$.

A process $A$ has finite variation on compact intervals when every sample path has finite total variation of a function there. Such a process can be integrated pathwise by the Lebesgue-Stieltjes integral.

Finite total variation of a function on $\mathbb{R}$ gives the following error bound for averaging on dyadic cells of length $\delta =2^{-j}$:The cell error is at most its length times the oscillation on that cell, and the sum of the oscillations is at most the total variation. This proves that the difference is integrable even if $f$ is not globally in $L^1$. For an even function decreasing to zero on the positive half-line, $\mathrm{TV}(f)\le 2f(0)$ and the bound becomes $f(0)2^{1-j}$.

A continuous adapted process $M$ is a local martingale if there are stopping times ${\tau }_n\uparrow \infty$ almost surely such that each stopped process $M_t^{{\tau }_n}=M_{t\wedge {\tau }_n}$ is a martingale. In this definition $M_0$ is integrable. A continuous adapted process $A$ has finite variation if, almost surely, its paths have finite total variation of a function on each compact time interval:The permitted fact that this total-variation process is continuous and adapted makes its level-hitting times stopping times.

Suppose $M=A$, and put $N=M-M_0$. DefineThese times increase to infinity almost surely. The optional stopping theorem applied to the localized martingale shows that $N^{{\sigma }_n}$ is a martingale; it is also bounded in absolute value by $n$, with total variation at most $n$. Fix $n,t$ and write $L=N^{{\sigma }_n}$. For a deterministic partition of an interval $0=t_0<\cdots <t_r=t$, orthogonality of square-integrable martingale increments givesPathwise the sum on the right is bounded by $n{\max }_j|L_{t_j}-L_{t_{j-1}}|$. For partitions with mesh tending to zero, continuity makes this converge to zero. It is bounded by $2n^2$, so dominated convergence gives $\mathbb{E}{L}_t^2=0$. First use rational $t$, then continuity, to obtain $L\equiv 0$ almost surely. Letting $n\to \infty$ provesThis proves continuous finite-variation local martingale is constant without assuming integrability of the unstopped total variation. Continuity matters: compensated jump martingales can have finite variation without being constant.

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.

Write ${\delta }_n=2^{-n}$ and let $Q_n(U)$ denote the printed squared-increment sum for a process $U$, on grid times $k{\delta }_n$. We construct its limit first for bounded martingales, then use localizing sequences and the finite variation part of a semimartingale.

Continuity and monotonicity of the bounded-martingale limit. Let $U$ be a uniformly bounded continuous martingale. The given result supplies a limit $R^U$ in uniform convergence on compacts in probability. Each $Q_n(U)$ is continuous and adapted. A subsequence converges almost surely uniformly on each compact time interval, by choosing summable error probabilities and diagonalizing. Its limit therefore has a continuous version; in the usual completed filtration this version is adapted.

The sums $Q_n(U)$ themselves can decrease between grid points, because the last partial increment is being squared. To establish monotonicity, use instead the quadratic variation from completed grid incrementsThese step processes are nondecreasing, and pathwise uniform continuity giveswhere ${\omega }_U(\delta ;T)=\sup \{|U_t-U_s|:s,t\le T,|t-s|\le \delta \}$. Thus the same almost surely uniform subsequence of $S_n$ converges to $R^U$, proving that $R^U$ is nondecreasing. Also $R_0^U=0$.

Localization of a continuous local martingale. Subtract the initial value, which does not affect increments, so that $M_0=0$. SetContinuity gives ${\tau }_m\uparrow \infty$ almost surely and makes $M^{{\tau }_m}$ bounded. The bounded local martingale criterion makes it a true martingale. Let $R^m$ be its continuous, adapted, nondecreasing limit.

The discrete sums commute exactly with stopping:For $\ell \le m$, uniqueness of the probability limit and stability of uniform convergence on compacts in probability under stopping giveup to indistinguishability. Taking a common null set for the countably many pairs, patch these processes into $R^M$ by setting $R_t^M=R_t^m$ when $t\le {\tau }_m$. Compatibility makes this definition independent of $m$. It is continuous, adapted, and nondecreasing. For each $T,\epsilon >0$,Let $n\to \infty$ and then $m\to \infty$. This proves the localization and patching of quadratic variation.

Adding finite variation. Decompose the continuous semimartingale as $X=X_0+M+A$, where $M$ is a continuous local martingale starting at zero and $A$ is continuous, adapted, and locally of finite variation. For every compact interval, the sum of the absolute increments of $A$ is at most its total variation of a function. Hence, pathwise,The Cauchy-Schwarz inequality bounds the mixed increment sum byThe first factor is bounded in probability, since $Q_n(M)\to R^M$ uniformly on compacts in probability; the second tends to zero almost surely. Thus the mixed term tends to zero in probability. Expanding the square provesThe constructed $R=R^M$ is continuous, nondecreasing, and adapted. This is the quadratic variation of $X$, and expresses the fact that finite-variation terms do not change quadratic variation.

Every dyadic partition is among the finite partitions on the right-hand side, so the displayed supremum is at least $\Vert f\Vert$. For the converse, the claim is immediate if $\Vert f\Vert =\infty$. If it is finite, apply part (a) to every interval of an arbitrary partition $0\le t_0<\cdots <t_n=1$:Summing telescopes and givesTaking the supremum proves that the dyadic definition equals the usual total variation of a function.

On bounded continuously differentiable functions with uniformly bounded total variation of a function, the map $T$ is Hadamard differentiable in the supremum norm with derivative ${\dot{T}}_{F,G}(a,b)={\int }_0^{u}b{F}^{\prime }+G(u)a(u)-G(0)a(0)-{\int }_0^u{G}^{\prime }a$. Use weak convergence of bounded-variation integrators for the first term and integration by parts for the remaining terms. No derivatives of the limiting continuous directions are required.

If continuously differentiable $F_j,F$ converge uniformly on a compact interval and have uniformly bounded total variation of a function, then $\int b{F}_j^{\prime }\to \int b{F}^{\prime }$ for every continuous $b$. For continuously differentiable test functions this is integration by parts. Uniform approximation by polynomials extends the conclusion to continuous tests using the variation bound.