Quadratic covariation of Brownian Ito processes

Finite-variation estimate: let $A_t={\int }_0^t{a}_{s}ds$ be pathwise absolutely continuous and $Y$ continuous. For a partition of $[0,T]$, [F9] gives ${\sum }_j\mathrm{\mid }{A}_{s_{j+1}}-A_{s_j}\mathrm{\mid }\le {\int }_0^T\mathrm{\mid }{a}_s\mathrm{\mid }ds$ and ${\max }_j\mathrm{\mid }{Y}_{s_{j+1}}-Y_{s_j}\mathrm{\mid }\le {\max }_j{\sup }_{u,v\in I_j}\mathrm{\mid }{Y}_u-Y_v\mathrm{\mid }\to 0$ along vanishing meshes, so $\mid {\sum }_j(A_{s_{j+1}}-A_{s_j})(Y_{s_{j+1}}-Y_{s_j})\mid \le ({\max }_j{\sup }_{u,v\in I_j}\mathrm{\mid }{Y}_u-Y_v\mathrm{\mid }){\int }_0^T\mathrm{\mid }{a}_s\mathrm{\mid }ds\to 0$ almost surely; the partial-increment convention obeys the same bound, so the cross sums of $(A,Y)$ converge to $0$ for every admissible sequence and $[A,Y]=0$.

Bilinearity: for continuous $X_1,X_2,Y$ whose displayed covariations exist, the partial sums satisfy ${\sum }_j\mathrm{\Delta }(X_1+X_2{)}_j\mathrm{\Delta }{Y}_j={\sum }_j\mathrm{\Delta }{X}_{1,j}\mathrm{\Delta }{Y}_j+{\sum }_j\mathrm{\Delta }{X}_{2,j}\mathrm{\Delta }{Y}_j$ exactly, and $[c{X}_1,Y]=c[X_1,Y]$ likewise; passing to the common probability limit gives $[X_1+X_2,Y]=[X_1,Y]+[X_2,Y]$ and $[c{X}_1,Y]=c[X_1,Y]$. By induction the same holds for finite sums, and by symmetry also in the second slot.

Same coordinate, general integrands: let $H,K$ be locally square-integrable predictable and $M_H=\int HdB$, $M_K=\int KdB$. For every interval $I=(u,v]$ of a partition, linearity of the integral gives ${\int }_I(H+K)dB={\int }_{I}HdB+{\int }_{I}KdB$, hence the exact identity $\mathrm{\Delta }{M}_H\mathrm{\Delta }{M}_K=\frac{1}{2}[(\mathrm{\Delta }{M}_H+\mathrm{\Delta }{M}_K{)}^2-(\mathrm{\Delta }{M}_H{)}^2-(\mathrm{\Delta }{M}_K{)}^2]$ with $\mathrm{\Delta }{M}_H+\mathrm{\Delta }{M}_K={\int }_I(H+K)dB$. Summing over the partition and applying the scalar quadratic-variation limit [F3] to the three locally square-integrable integrands $H+K,H,K$ yields convergence in probability, uniformly in $t$, of the cross sums to $\frac{1}{2}[{\int }_0^t(H+K{)}^{2}ds-{\int }_0^t{H}^{2}ds-{\int }_0^t{K}^{2}ds]={\int }_0^{t}HKds$, for the step convention; the partial-increment convention has the same limit by [F3]. Hence $[\int HdB,\int KdB{]}_t={\int }_0^{t}HKds$ for locally square-integrable predictable $H,K$.

Distinct coordinates, bounded elementary integrands: take $k\mathrm{\ne }l$, and a common elementary coefficient partition for $H,K$ with $L$ blocks and an a.s. deterministic coefficient bound $C$. Refine ${\pi }_n$ by that fixed partition, obtaining $0=r_0<\cdots <r_J=T$ with mesh at most ${\delta }_n$. On $(r_j,r_{j+1}]$ the coefficients ${\alpha }_j,{\beta }_j$ are ${\mathcal{F}}_{r_j}$-measurable and bounded by $C$. The refined cross sum has increments $D_j={\alpha }_j{\beta }_j(B_{r_{j+1}}^k-B_{r_j}^k)(B_{r_{j+1}}^l-B_{r_j}^l)$, $0\le j<J$. By [F6], their conditional means given ${\mathcal{F}}_{r_j}$ vanish and $E{D}_j^2\le C^4(r_{j+1}-r_j{)}^2$. Thus $S_q={\sum }_{0\le j<q}{D}_j$, $0\le q\le J$, is a square-integrable martingale with $S_0=0$. Orthogonality and discrete Doob, extending it constantly after $J$ if necessary, give $E{\max }_{q\le J}\mathrm{\mid }{S}_q{\mathrm{\mid }}^2\le 4C^4{\delta }_{n}T$. Therefore the refined step cross sums tend uniformly to zero in probability. The coefficients may depend on both Brownian coordinates; only the vector increment's independence of the past is used.

For any continuous pair of integral paths $U,V$, the partial-increment cross sum differs from the step cross sum by $(U_t-U_{s_{j(t)}})(V_t-V_{s_{j(t)}})$ on its final interval. Its supremum is bounded by ${\omega }_U({\delta }_n){\omega }_V({\delta }_n)\to 0$ on their common continuity event. The measurable-supremum convention of [F2] makes this an almost-sure error bound and therefore an error tending to zero in probability.

Refinement discrepancy: for sufficiently large $n$, a ${\pi }_n$ interval crosses at most one fixed coefficient boundary. Let ${\omega }_k(\delta )$ and ${\omega }_l(\delta )$ be the path moduli of the two Brownian coordinates on $[0,T]$. On a crossing interval, each original integral increment has absolute value at most $2C$ times its Brownian modulus, so the original cross product is bounded by $4C^2{\omega }_k({\delta }_n){\omega }_l({\delta }_n)$. The two refined cross products together are bounded by $2C^2{\omega }_k({\delta }_n){\omega }_l({\delta }_n)$. This also covers an intermediate time when only one refined increment has been completed. Hence the absolute difference between the original and refined step cross sums is uniformly bounded by $6L{C}^2{\omega }_k({\delta }_n){\omega }_l({\delta }_n)$, which tends to zero almost surely by [F9]. This estimate uses interval oscillations, not the absolute full-interval increments, which could cancel.

By steps 1.4 and 2.1 and the triangle/union bound, the original step cross sums for bounded elementary $H,K$ in distinct coordinates tend uniformly to zero in probability. Step 1.5 proves the same for partial-increment sums. Thus the two integral processes have zero covariation for every deterministic vanishing-mesh partition sequence.

Finite-energy approximation: write $C_n(H,K)(t)$ for the step cross sum of the two integral processes. Choose bounded elementary $H^{\mathrm{\prime }},K^{\mathrm{\prime }}$ with respective $L^2(\mathrm{d}t\otimes P)$ errors at most $\eta >0$, using [F5] and [F10]. Bilinearity gives $C_n(H,K)-C_n(H^{\mathrm{\prime }},K^{\mathrm{\prime }})=C_n(H-H^{\mathrm{\prime }},K)+C_n(H^{\mathrm{\prime }},K-K^{\mathrm{\prime }})$. For either term, finite-sum Cauchy–Schwarz bounds the supremum over completed partial sums by the product of the two terminal quadratic sums' square roots. Taking expectation and using [F4], [F8] bounds the expected supremum of the difference by $\eta \mathrm{\parallel }K{\mathrm{\parallel }}_2+(\mathrm{\parallel }H{\mathrm{\parallel }}_2+\eta )\eta$, uniformly in $n$. This also covers $K=0$ and needs no bound of the form $\mathrm{\parallel }{K}^{\mathrm{\prime }}{\mathrm{\parallel }}_2^2\le 2\mathrm{\parallel }K{\mathrm{\parallel }}_2^2$. The nonnegative indicator inequality bounds the approximation probability by this expectation divided by its error threshold. At fixed $\eta$ the elementary error vanishes by step 3.1; then let $\eta \downarrow 0$. Step 1.5 handles partial increments. Thus distinct-coordinate covariation vanishes for finite-energy predictable integrands.

Localization: let ${\tau }_N^H,{\tau }_N^K$ be the respective canonical energy stopping times, $N\ge 1$, and put ${\rho }_N={\tau }_N^H\wedge {\tau }_N^K$. These are nondecreasing stopping times tending to infinity almost surely, and the expected energies of both $H{1}_{(0,{\rho }_N]}$ and $K{1}_{(0,{\rho }_N]}$ are at most $N$. By [F4], on $\{{\rho }_N\ge T\}$ and a common probability-one agreement event the two original integrals equal their finite-energy stopped integrals on $[0,T]$. For any error threshold $\epsilon$, the original cross-sum error probability is at most $P({\rho }_N<T)$ plus the stopped cross-sum error probability. The latter tends to zero by step 4.1 for fixed $N$; the former tends to zero as $N\to \mathrm{\infty }$. No bound on the cross sum on the exceptional event is needed. This proves zero covariation for locally square-integrable integrands in distinct coordinates, for both conventions. Positive indices are reindexed by $N=j+1$ when required.

General matrix: $M^i={\sum }_k{M}^{ik}$ and $M^j={\sum }_l{M}^{jl}$ are finite sums; by the bilinearity of step 1.2 applied repeatedly, and the existence of each pairwise covariation from steps 1.3 and 5.1, $[M^i,M^j]={\sum }_{k,l}[M^{ik},M^{jl}]={\sum }_k{\int }_0^{\cdot }{\sigma }_s^{ik}{\sigma }_s^{jk}ds+{\sum }_{k\mathrm{\ne }l}0$, where the second sum is zero by step 5.1 applied to the locally square-integrable integrands ${\sigma }^{ik}$ and ${\sigma }^{jl}$; hence $[M^i,M^j{]}_t={\int }_0^t(\sigma {\sigma }^{\mathsf{T}}{)}_s^{ij}ds$. Every step was proved for an arbitrary deterministic vanishing-mesh sequence and for both conventions, so the existence clause of [F2] is met.

Drift parts contribute zero: the drift processes $A_t^i={\int }_0^t{b}_s^{i}ds$ are pathwise absolutely continuous, so step 1.1 with $[F1]$ gives $[A^i,A^j]=0$ and $[A^i,Y]=0$ for every continuous $Y$, in particular for $Y=M^j$ and for $Y=B^k$. Therefore, using bilinearity [step 1.2] and $X^i=X_0^i+A^i+M^i$, $[X^i,X^j]=[A^i,A^j]+[A^i,M^j]+[M^i,A^j]+[M^i,M^j]=[M^i,M^j]={\int }_0^{\cdot }(\sigma {\sigma }^{\mathsf{T}}{)}_s^{ij}ds$. This proves clause 2 of the statement for all continuous $Y$, since the estimate of step 1.1 applies to an arbitrary continuous second factor.

Boundary and consistency cases: for $d=1$ and $m=1$ the formula reads $[X{]}_t={\int }_0^t{\sigma }_s^{2}ds$, and taking $H=K=1$ in step 1.3 recovers the Brownian identity $[B,B{]}_t=t$; if $\sigma \equiv 0$, only the drift remains and its covariation is zero; if $b\equiv 0$, only the drift term vanishes and the stochastic covariation generally remains nonzero (Brownian motion is the simplest example); at $t=0$ both sides vanish, since empty cross sums and empty integrals are $0$; and for a degenerate $d$-dimensional process with singular dispersion matrix the formula still holds matrix-wise, with no independence of the coordinates assumed. AC enters only through [F10], which supplies the approximations and versions; the energy stopping times are canonical.