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For a left-continuous locally bounded adapted process $H$ and a continuous semimartingale $X$, the elementary predictable left-endpoint approximations tend pointwise to $H$. Decompose $X$ into a continuous local martingale and a continuous finite-variation process. After localization, dominated convergence with respect to the variation measure handles the finite-variation integral. The Itô isometry and Doob L2 maximal inequality handle the martingale integral. The omitted partial last interval is bounded by the local bound on $H$ times the modulus of continuity of $X$. This proves uniform convergence on compacts in probability and is the grid-sum foundation of the Itô product rule.

Stop a continuous local martingale at increasing level-and-time stopping times to obtain bounded martingales. Their limits of discrete quadratic variation sums agree before the earlier stopping time, because the sums commute exactly with stopping. These continuous limits patch into a continuous adapted nondecreasing process. On each compact interval, the chance that the stopping time occurs early tends to zero, giving uniform convergence on compacts in probability for the original sums.

Fix a sequence of deterministic partitions with mesh tending to zero and define the completed-increment sumsFor a continuous semimartingale, these sums converge to $[X{]}_t$ in uniform convergence on compacts in probability, the standard path-increment formula for quadratic variation. The sums use only the path of $X$, and do not mention a filtration. If the same process is a semimartingale for two filtrations, the same sums have both proposed quadratic variations as limits. Uniqueness of limits in probability, followed by continuity, makes these limits indistinguishable.

For the change of measure, let $R=d\mathbb{Q}/d\mathbb{P}$. If $\mathbb{P}(A_n)\to 0$, thentruncate $R$ at a large constant, use $\mathbb{E}[R{1}_{\{R>K\}}]\to 0$, and then let $\mathbb{P}(A_n)\to 0$. Apply this argument to the event ${\sup }_{s\le t}|S_s^{(n)}-[X{]}_s^{\mathbb{P}}|>\epsilon$. The same sums converge to the P-version of the bracket in Q-probability, and the Q-semimartingale formula makes them converge to the Q-version. ThereforeThis is quadratic variation under an absolutely continuous measure change. With only $\mathbb{Q}\ll \mathbb{P}$, the equality is asserted under Q: a Q-version is unconstrained on a Q-null set that might have positive P-probability.

For continuous semimartingales $U,V$, the semimartingale integration by parts formula isIndeed each partition increment satisfiesSumming telescopes the left side. The first two sums converge to the stochastic integrals by left-endpoint approximation, and the last sum converges to the quadratic covariation by polarization of the quadratic-increment formula. All limits are in uniform convergence on compacts in probability. Hence the equality holds at every time outside one null set, using continuity. In differential notation it is the Itô product rule $d(UV)=UdV+VdU+d[U,V]$.

For a vector of continuous semimartingales $X=(X^1,\dots ,X^d)$ and $F\in C^{1,2}$, the multidimensional Itô formula isOne route from the product formula is to apply it repeatedly to powers and products of the coordinates, obtaining this expression for polynomials; polarization identifies the mixed second derivatives. Time is another coordinate of finite variation, so it has no quadratic-covariation contribution. Localize to compact ranges, approximate a $C^{1,2}$ function and its indicated derivatives by smooth polynomial approximations, and pass to the limit using stochastic-integral continuity and the variation bounds on the finite-variation terms. Equivalently, second-order Taylor expansion along partitions gives the same limit, with integration by parts accounting for every quadratic term.

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.

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.

A left-continuous adapted process is a predictable process. Define the elementary predictable approximationAt every $s>0$, its sample time approaches $s$ from the left, including when $s$ is itself a grid point. Thus $H_s^{(n)}\to H_s$ pathwise by left continuity.

Use the continuous semimartingale decomposition $X=X_0+M+A$, with $M$ a continuous local martingale and $A$ a continuous finite-variation process. Localize so that on a fixed horizon $T$, the integrands are bounded by a deterministic constant, $[M{]}_T$ is bounded, and the total-variation process of $A$ at $T$ is bounded. This is possible using local boundedness and the finite pathwise clock and variation.

For the finite-variation integral, the dominated convergence theorem givesalmost surely. For the martingale integral, the Itô isometry and the Doob L2 maximal inequality giveHere the localized bounds justify dominated convergence also in expectation. Remove localization to conclude that the full elementary integrals converge in uniform convergence on compacts in probability to $\int HdX$.

The sum of completed intervals differs from the full elementary integral at time $t$ byIts maximum on $[0,T]$ tends to zero almost surely, by the local bound on $H$ and uniform continuity of $X$. ThereforeThis proves left-endpoint approximation of a continuous semimartingale integral, including the unfinished interval convention.

Define the quadratic covariation of two continuous semimartingales by polarization:The available quadratic variation convergence theorem and the polarization identity imply that the sums of products of simultaneous grid increments converge in uniform convergence on compacts in probability to $[X,Y]$. The quadratic covariation is a continuous finite-variation process, vanishing at time zero.

For a grid ending at $t_n=h_n\lfloor t/h_n\rfloor$, the elementary algebraic identitytelescopes. The first two sums converge by part (a), with the continuous processes $X,Y$ as integrands; continuous paths are left-continuous and locally bounded. The last sum converges to the quadratic covariation by the preceding polarization argument. Finally, $X_{t_n}{Y}_{t_n}\to X_t{Y}_t$ locally uniformly by continuity. HenceThis is the Itô product rule, or stochastic integration by parts. Uniqueness of the limit in uniform convergence on compacts in probability, followed by continuity, makes the identity hold simultaneously for all $t$ outside one null set. The proof used finite-grid algebra, part (a), and the permitted quadratic variation theorem; it did not use the Itô formula.

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.

Put $M_k=S_k-mk$ as above. Centering the linear interpolation also interpolates the centered values: for $u\in [0,1]$,The absolute value of this convex combination is at most the larger endpoint absolute value. For every fixed $A>0$, part (c) therefore givesHence there is uniform convergence on compacts in probability to the deterministic path $h(t)=mt$, by Markov inequality. Equip $C([0,\infty ),\mathbb{R})$ with its standard compact-open topology, metrized byFor each finite number of terms their suprema converge to zero in probability, and the remaining tail is bounded deterministically by ${\sum }_{j>J}{2}^{-j}$. Thus $d(S^{(N)},h)\to 0$ in probability. Convergence in probability to a deterministic point implies weak convergence of probability measures, so the fluid limit isHere ${\delta }_h$ is the Dirac measure concentrated on that continuous path. The topology is locally uniform convergence; no assertion of uniform convergence over the entire unbounded half-line is needed. The interpolation is intended for integers $k\ge 0$, including the initial interval $[0,1]$. If the PDF's ${\mathbb{Z}}^{+}$ is interpreted as strictly positive integers, that initial piece is omitted from the displayed definition and must be supplied by the same formula using $S_0=0$.

The relevant fact is uniform convergence on compacts in probability under an absolutely continuous measure change. Write $Z=d\tilde{\mathbb{P}}/d\mathbb{P}$. For any event $E$ and $K>0$,The last term tends to zero as $K\to \infty$ because $Z$ is integrable. Consequently, $\mathbb{P}(E_n)\to 0$ implies $\tilde{\mathbb{P}}(E_n)\to 0$. Apply this to $E_n=\{{\sup }_{t\le T}|C_t^n-[M,N{]}_t|>\epsilon \}$ for each finite $T$ and $\epsilon >0$. The given dyadic sums therefore have the same uniform convergence on compacts in probability limit under the new probability measure.

This proves quadratic covariation under an absolutely continuous measure change:Taking $N=M$ proves the corresponding quadratic variation assertion directly; it has not been assumed. The ceiling in the printed sum includes one grid interval extending beyond $t$. This does not affect the limit: on $[0,T]$ the resulting extra product is bounded by the product of the two path oscillations on mesh $2^{-n}$, which tends to zero by uniform continuity on $[0,T+1]$.

The bracket under the new probability measure must be understood as the quadratic covariation of continuous semimartingales. Semimartingale stability under an absolutely continuous measure change ensures this class is preserved; the local martingale property need not be. For example, on a finite horizon, weighting Brownian motion by $\exp (\mu B_T-{\mu }^{2}T/2)$ makes $B_t-\mu t$ a Brownian motion by the Girsanov theorem, so $B$ has nonzero drift under the new measure even though its quadratic variation is still $t$. The PDF additionally assumes that $M,N$ remain local martingales under the new probability measure, so its bracket is also defined directly by the local martingale characterization.

The limit is in probability, or more strongly in uniform convergence on compacts in probability; a pathwise limit for every continuous local martingale is not implicit. The original PDF uses $\lceil t{2}^n\rceil$, whereas the supplied TeX changes this to $\lfloor t{2}^n\rfloor$; both versions have the same limit by continuity.

Put $\delta =2^{-n}$ and $m=\lfloor t/\delta \rfloor$. For the completed intervals, the algebraic identityseparates a left-point sum from a quadratic covariation sum.

The results from stochastic calculus used are these: predictable left-step approximations to a continuous adapted integrand converge in their Itô integrals in uniform convergence on compacts in probability; and products of increments of continuous semimartingales on deterministic partitions of an interval with vanishing mesh converge to their quadratic covariation in the same sense. The first follows after localization from the Itô isometry and the Doob L2 maximal inequality, since the integrand approximations converge uniformly on compact intervals; the second is the deterministic-partition characterization of quadratic covariation.

Thus the two sums converge to ${\int }_0^t{Y}_{s}d{X}_s$ and $\frac{1}{2}⟨Y,X{⟩}_t$. For the ceiling convention, at most one additional full interval extends past $t$; on $[0,T+1]$ its absolute contribution is at most ${\sup }_{s\le T+1}|Y_s|$ times the modulus of continuity of $X$ at scale $\delta$, which tends to zero almost surely. Therefore

A semimartingale is an adapted process of the form $X=X_0+M+A$, where $M$ is a local martingale and $A$ is an adapted finite-variation process. A sequence of processes $X^n$ converges to $X$ in uniform convergence on compacts in probability, abbreviated ucp, when for every $T<\infty$ and $\delta >0$,

A càdlàg process $A$ is a finite-variation process when, almost surely, for every $T<\infty$,where the supremum is over every finite partition of an interval $0=t_0<\cdots <t_n=T$.

Uniform convergence on compacts in probability of $X^n$ to $X$ means that, for every $T<\infty$ and $\epsilon >0$,

Apply the realized absolute covariation theorem to each dyadic partition of an interval. More explicitly, use the continuous increasing clock $C=[M]+[N]$ and the Radon-Nikodym theorem to writeFor each time, let $(U,V)$ have the centered bivariate normal distribution with covariance matrix $\left(\begin{array}{cc}a& c\\ c& b\end{array}\right)$, and defineThis process is continuous and increasing. To prove convergence, localize $M,N$, represent the pair as stochastic integrals against a two-dimensional Brownian motion, and approximate the integrands in $L^2(dC)$ by bounded step previsible processes. For step integrands, the result is the weak law of large numbers applied on each block to independent Gaussian random variables. The Burkholder-Davis-Gundy inequality and the Cauchy-Schwarz inequality make the error uniform on each compact interval in probability. Consequentlyin the sense of uniform convergence on compacts in probability.

Let ${\tau }_m=\inf \{s:|X_s|\ge m\}$. The stopped process $X^{{\tau }_m}$ is bounded, so part (a) gives a continuous adapted quadratic variation $A^{[m]}$. These processes agree before the smaller stopping time, because their dyadic sums agree there and the limits are unique in probability. They therefore paste to a continuous adapted process $A$ with $A_{s\wedge {\tau }_m}=A_s^{[m]}$.

For fixed $t$ and $\epsilon >0$,The second term tends to zero by part (a), while continuity of $X$ on $[0,t]$ makes $\mathbb{P}({\tau }_m\le t)\to 0$. This proves convergence uniformly on compact intervals in probability.

Fix a deterministic horizon $T$ and $\epsilon >0$. For every localization index $N$,by Markov inequality. Because $T_N\uparrow \infty$ almost surely, the first term tends to zero as $N\to \infty$. For fixed $N$, the second tends to zero as $n\to \infty$. Taking first the limit superior in $n$ and then $N\to \infty$ provesin probability. This is precisely uniform convergence on compacts in probability.