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For a continuous vector semimartingale $X$ and $f\in C^{1,2}$, the Itô formula isThe Itô product rule, or stochastic integration by parts, isThe additional second-order term is the quadratic covariation.

For general semimartingales with càdlàg paths, use left limits in the integrands. The Itô formula for semimartingales with jumps, for time-independent $f\in C^2$, isHere $X^c$ denotes the continuous local martingale part; the continuous bracket is that of these parts. For time-dependent $f\in C^{1,2}$ add $\int {\partial }_{s}f(s,X_{s-})ds$ and insert $s$ in the other evaluations of $f$. The integration by parts formula becomesFor continuous paths the jump sum vanishes, recovering the first two formulas.

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.

For continuous semimartingales, the Stratonovich integral adds half the quadratic covariation to the Itô integral:One can see the factor directly from symmetric endpoint sums. On a partition of an interval, replacing $X$ at the left endpoint by the average of its two endpoint values adds $\frac{1}{2}\sum \Delta X\Delta Y$; its limit is $\frac{1}{2}[X,Y]$. Thus the correction depends only on the continuous local martingale parts of the semimartingales. If either integrator or integrand has finite variation, the quadratic covariation vanishes and these two integrals coincide. Here the printed notation $\partial Y$ denotes Stratonovich integral integration, equivalently $\circ dY$.

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

Interpret the Stratonovich integral in its standard extension to continuous semimartingale integrands:This extension is needed because $f^{\prime }(X)$ is generally a semimartingale rather than a local martingale. Also the introductory prose in the PDF reverses integrand and integrator relative to its displayed definition; the displayed definition fixes the convention used here.

Apply the Itô formula to $f^{\prime }(X)$, using $f\in C^3$:The continuous finite-variation process in the second term has zero quadratic covariation with $X$, while the quadratic covariation rule for an Itô integral givesA second application of the Itô formula, now to $f(X)$, therefore yieldsAll unbounded coefficients are handled by stopping $X$ on compact spatial intervals, then removing the localizing sequence.

Let ${\theta }_t$ and ${\eta }_t$ be the predictable holdings of the stock and the bank account. With no consumption, the self-financing portfolio has $X=\theta S+\eta B$ and $dX=\theta dS+\eta dB$. The Itô product rule, together with the dynamics from part (a), givesThe drift vanishes because $X=\theta S+\eta B$ and $\sigma \lambda =\mu -r$. Thus $XY$ is a local martingale. It is nonnegative by the assumed nonnegative wealth and strict positivity of $Y$, and henceFor example, this last fact follows directly from Conditional Fatou lemma applied to a localizing sequence; the initial capital is assumed finite. As usual, holdings must be integrable against the asset semimartingales for the self-financing portfolio equation to be defined.

Fix $z$ and writeBy assumption, $L(z)$ is a continuous local martingale, so $Z_t(z)=e^{L_t(z)}$ is a semimartingale. The Chordal Loewner equation giveswhich has finite variation. Thereforeis a semimartingale. Thus the Loewner driver $U$ is a continuous semimartingale.

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 $\delta >0$,

For continuous semimartingales and $Q\ll P$, their quadratic covariation has the same version under the two probability measures, up to $Q$-indistinguishability of stochastic processes. The dyadic product sums have the same limit by uniform convergence on compacts in probability under an absolutely continuous measure change. Uniqueness of a limit in probability identifies that limit with the bracket under $Q$. Semimartingale stability under an absolutely continuous measure change supplies the correct class under $Q$: a local martingale need not remain a local martingale. Taking the two processes equal proves the same statement for quadratic variation without a separate assumption.

If $Q\ll P$ and a continuous semimartingale is a semimartingale under both measures, its quadratic variations agree $Q$-indistinguishably. The same squared-increment sums converge uniformly on compacts in probability under both measures: absolute continuity transfers the original convergence, and uniqueness of the limit identifies the two continuous versions. Equivalence of measures is unnecessary.

On a fixed finite horizon, equivalent probability measures have the same semimartingales. For a strictly positive continuous Radon-Nikodym density martingale $Z$ and a continuous local martingale $L$ under the original measure, $L-\int Z^{-1}d[L,Z]$ is a local martingale under the new one, by the Girsanov theorem. Thus the finite-variation part of a continuous semimartingale decomposition changes by the opposite covariation correction.

If $X$ is a semimartingale under $P$ and $Q\ll P$, then it is a semimartingale under $Q$. The Bichteler-Dellacherie theorem gives a short proof: terminal elementary integrals associated with uniformly small elementary predictable processes with stopping-time intervals converge to zero in $P$-probability. Uniform convergence on compacts in probability under an absolutely continuous measure change, or its single-random-variable proof, transfers convergence to $Q$. The same good-integrator characterization now applies under $Q$. Adaptedness and càdlàg paths are preserved because $P$-null sets remain $Q$-null; if necessary use the usual $Q$-completion of the same filtration. If the integrands are bounded only outside $Q$-null sets, clip their coefficients to the specified deterministic bounds before applying the $P$-criterion; this preserves their $Q$-versions. Completion adds only equivalent versions of measurable coefficients. This theorem does not preserve the local martingale property.

Stochastic calculus extends integration and differential calculus to stochastic processes such as Brownian motion and semimartingales.

For a continuous semimartingale $X$ and a sufficiently smooth $f$, the Stratonovich integral obeys $f(X_t)-f(X_0)={\int }_0^t{f}^{\prime }(X_s)\circ d{X}_s$. In general $f^{\prime }(X)$ is a semimartingale, so the definition must allow this larger class of integrands.

For continuous semimartingales, the Stratonovich integral is defined by ${\int }_0^t{Y}_s\circ d{X}_s={\int }_0^t{Y}_{s}d{X}_s+\frac{1}{2}[Y,X{]}_t$, where the first term is an Itô integral and the correction is quadratic covariation. It obeys the ordinary chain rule and is the limit in probability of symmetric endpoint sums.