Quadratic covariation of Brownian Ito processes

Fix processes $X=(X_t{)}_{t\ge 0}$ and $Y=(Y_t{)}_{t\ge 0}$ of real random variables on one probability space $(\mathrm{\Omega },\mathcal{F},P)$: for every $\omega$ in a measurable event of probability one the paths $t\mapsto X_t$ and $t\mapsto Y_t$ are continuous on $[0,\mathrm{\infty })$ Continuity of $f:A\to \mathbb{R}$ at a point of $A$ and on $A$: the $\epsilon$-$\delta$ condition, its agreement with ${\lim }_{x\to c}f(x)=f(c)$ at a limit point, and continuity at an isolated point. Fix $T>0$ and a deterministic partition sequence $({\pi }_n)$ of $[0,T]$ with mesh tending to $0$ Quadratic variation along a partition sequence. For each $n$ write ${\pi }_n=(m_n,s^{(n)})$ and form the two families of cross-increment partial sums $S_n^{\mathrm{s}\mathrm{t}\mathrm{e}\mathrm{p}}(t):={\sum }_{\begin{array}{c}1\le k\le m_n\\ s_k^{(n)}\le t\end{array}}(X_{s_k^{(n)}}-X_{s_{k-1}^{(n)}})(Y_{s_k^{(n)}}-Y_{s_{k-1}^{(n)}}),$ $S_n^{\mathrm{p}\mathrm{a}\mathrm{r}\mathrm{t}}(t):=S_n^{\mathrm{s}\mathrm{t}\mathrm{e}\mathrm{p}}(t)+(X_t-X_{s_{k(t)}^{(n)}})(Y_t-Y_{s_{k(t)}^{(n)}})(k(t)<m_n),$ with $S_n^{\mathrm{p}\mathrm{a}\mathrm{r}\mathrm{t}}(T):=S_n^{\mathrm{s}\mathrm{t}\mathrm{e}\mathrm{p}}(T)$ and $k(t)$ the largest index in $\{0,\dots ,m_n\}$ with $s_{k(t)}^{(n)}\le t$; these are the cross sums corresponding to the two conventions of Quadratic variation along a partition sequence specialized to the pair $(X,Y)$, and both are $0$ at $t=0$.

All suprema in probability statements below use measurable versions. For any finite collection of the processes involved, including a candidate limit, intersect their measurable probability-one continuity events and set all of them to zero off that intersection. These representatives have everywhere continuous paths and retain measurable fixed-time values. The partial-sum paths are continuous; the step-sum paths are right-continuous on $[0,T)$ with the specified value at $T$. Their uniform distances from a continuous candidate therefore equal suprema over $(\mathbb{Q}\cap [0,T])\cup \{T\}$, which are finite measurable random variables. Another such normalization agrees on a measurable probability-one event and gives the same probability limits. Here indistinguishability means agreement at every time on a measurable probability-one event. No completeness or adaptedness is needed for this convention.

1. Existence and value of the covariation. We say that the quadratic covariation $[X,Y]$ exists on $[0,T]$ when there is a real process $t\mapsto [X,Y{]}_t$ of real random variables with almost-sure continuous paths on $[0,T]$ such that for every deterministic partition sequence $({\pi }_n)$ of $[0,T]$ with mesh tending to $0$ both families converge to it uniformly in probability on $[0,T]$: ${\sup }_{0\le t\le T}\mid S_n^{\mathrm{s}\mathrm{t}\mathrm{e}\mathrm{p}}(t)-[X,Y{]}_t\mid ⟶0,{\sup }_{0\le t\le T}\mid S_n^{\mathrm{p}\mathrm{a}\mathrm{r}\mathrm{t}}(t)-[X,Y{]}_t\mid ⟶0,$ the convergence being convergence in probability Convergence in probability. The two displayed requirements are part of one condition: the same process $[X,Y]$ must arise for every admissible sequence and for both conventions. When the condition holds we call $[X,Y{]}_t$ the quadratic covariation of $X$ and $Y$ at time $t$, and we write $[X]:=[X,X]$ and call it the quadratic variation of $X$. For two candidate limits $U,V$, fix a deterministic dyadic partition sequence. The triangle inequality bounds ${\sup }_t\mathrm{\mid }{U}_t-V_t\mathrm{\mid }$ by the sum of their uniform errors against its partial sums. For each $\epsilon >0$ the probability that this supremum exceeds $\epsilon$ is at most the sum of the two error probabilities at $\epsilon \mathrm{/}2$, and hence is zero. Taking $\epsilon =1\mathrm{/}j$, $j\ge 1$, proves uniqueness on one measurable probability-one event; the definition is applied separately on each finite horizon, and when the covariations on all horizons are compatible we write the resulting process on $[0,\mathrm{\infty })$ again as $[X,Y]$.

2. Symmetry and polarization. Exchanging the two factors does not change the definition, so $[X,Y]=[Y,X]$ whenever either side exists. The identities $[X+Y]=[X]+2[X,Y]+[Y],[X-Y]=[X]-2[X,Y]+[Y]$ hold on the domain where all covariations appearing in them exist: the cross-increment sums are bilinear in the pair, so the displayed identities are exact at the level of partial sums for every partition, and probability limits pass through finite algebraic identities. In particular $[X,Y]=\frac{1}{2}([X+Y]-[X]-[Y])$ and $4[X,Y]=[X+Y]-[X-Y]$ on that domain.

3. Bilinearity and insensitivity to constants. If the covariations $[X_1,Y]$, $[X_2,Y]$ and $[X_1+X_2,Y]$ exist on $[0,T]$, then $[X_1+X_2,Y]=[X_1,Y]+[X_2,Y]$ there, and $[cX,Y]=c[X,Y]$ for real $c$; moreover $[X+c,Y]=[X,Y]$ for every real constant $c$, because the increments of a constant process are zero. These are again exact identities of partial sums plus uniqueness of limits.

4. Zero covariation with a continuous finite-variation process. Let $A=(A_t)$ be continuous on a full-measure event and suppose, for every $\omega$ in that event, that $t\mapsto A_t(\omega )$ has bounded variation on every finite interval in the sense of Bounded variation and total variation on an interval. Then for each $T$ and each admissible partition sequence, the cross sums of $A$ against any continuous process $Y$ satisfy $\mid {\sum }_k(A_{s_k}-A_{s_{k-1}})(Y_{s_k}-Y_{s_{k-1}})\mid \le ({\max }_k{\sup }_{u,v\in [s_{k-1},s_k]}\mathrm{\mid }{Y}_u-Y_v\mathrm{\mid }){\mathrm{Var}}_{[0,T]}(A),$ because every sum of absolute $A$-increments is bounded by the path's total variation on $[0,T]$; the maximum tends to $0$ along vanishing meshes by uniform continuity of the continuous path $Y$ on the compact interval $[0,T]$. Hence $[A,Y]$ exists and equals the zero process for every such $A$ and every continuous $Y$, and in particular $[A,A^{\mathrm{\prime }}]=0$ for two such processes. In the notation of the page this says that continuous finite-variation parts contribute nothing to quadratic covariation.

5. Scope and choice. The inputs and candidate limits have measurable fixed-time values and almost-sure continuous paths. The normalization above makes the uniform errors real random variables, as required by Convergence in probability. No filtration or adaptedness is used. The algebra in clauses 2--3 passes to limits by the uniform triangle inequality and the union bound. In clause 4 the same estimate holds for every partial sum, including its terminal partial increment, because its intervals are disjoint; it thus proves uniform convergence to zero on the common continuity event. For each epsilon, the measurable events that some error after index n exceeds epsilon decrease to a null event; continuity from above of probability gives convergence in probability. Uniform continuity is justified by the choice-free finite-cover argument in Quadratic variation along a partition sequence. The partitions are given and normalization uses a finite intersection of supplied full-measure events, so no choice axiom is used. Existence for Brownian Ito processes is a separate theorem, not a presupposition of this definition.