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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.

Because independent Brownian motions have zero quadratic covariation, the Itô product rule givesAfter integration, the random variable in the question is $B_t^1{B}_t^2$. Writing $B_t^j=\sqrt{t}{Z}_j$ for independent standard Gaussian random variables $Z_1,Z_2$, its distribution isthe scaled product of two independent standard normal random variables.

The martingale product identity says that $N_t{K}_t-[N,K{]}_t$ is a martingale. Passing to the terminal values of the square-integrable martingales and using $N_0=K_0=0$ givesThe quadratic covariation identity for a stochastic integral isApplying the same product identity to $H\cdot M$ and $K$ therefore gives

For an analytic function $f=u+iv$ and planar Brownian motion $Z=X+iY$ with independent standard coordinates, the Cauchy-Riemann equations make $u,v$ harmonic. The Itô formula gives $dU=u_{x}dX+u_{y}dY$ and $dV=-u_{y}dX+u_{x}dY$. Localizing the derivatives proves both coordinates are local martingales. Squaring their diffusion coefficients and multiplying the two rows gives the displayed quadratic variations and zero quadratic covariation.

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.

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.