Proving the Multidimensional Ito Formula

From Taylor Expansion to the Hessian Correction

For a smooth path \(x(t)\), the ordinary chain rule accounts for first-order changes in time and space. An Ito process has an additional second-order contribution: its random increments are of order \(\sqrt{\Delta t}\), so products of two such increments can accumulate to a nonzero limit over a partition. In several dimensions, these accumulated products include both squared increments and mixed products. The mixed terms are why the correction involves the full covariance matrix, not just the variances of the individual coordinates.

Work on a filtered probability space \((\Omega,\mathcal{F},(\mathcal{F}_t)_{t\ge0},\mathbb{P})\) carrying a \(d\)-dimensional standard Brownian motion \(\mathbf{B}_t=(B_t^1,\ldots,B_t^d)\). Let \(X_t=(X_t^1,\ldots,X_t^n)\) satisfy \(dX_t^i=a_i(t)\,dt+\sum_{k=1}^d\sigma_{ik}(t)\,dB_t^k\). Assume the coefficients are progressively measurable, \(\mathbb{E}\int_0^T(|a(t)|^2+\|\sigma(t)\|^2)\,dt<\infty\), and the derivatives used below are bounded. Thus each Brownian integrand in the formula belongs to \(\mathcal{L}^2\), the class of progressively measurable \(H\) with \(\mathbb{E}\int_0^T H_t^2\,dt<\infty\). The boundedness assumptions are convenient sufficient conditions; the usual local formula follows by stopping when the coefficients or derivatives leave bounded sets.

The matrix \(c(s)=\sigma(s)\sigma(s)^\mathsf{T}\) is the instantaneous covariance matrix of the coordinates of \(X\). In particular, \(c_{ii}\) gives the variance rate of coordinate \(i\), while \(c_{ij}\) for \(i\ne j\) gives the covariation rate between coordinates \(i\) and \(j\). The proof below shows where each entry appears.

Partition Proof and Remainder Control

Fix \(t\le T\) and take partitions \(0=t_0<t_1<\cdots<t_m=t\) whose mesh tends to zero. Write \(\Delta t_r=t_r-t_{r-1}\), \(\Delta X_r=X_{t_r}-X_{t_{r-1}}\), and evaluate derivatives with subscript \(r-1\) at \((t_{r-1},X_{t_{r-1}})\). Split each change into a time change at the new spatial point and a spatial change at the old time:

Continuity of \(f_t\) gives a first-order time increment \(f_t(t_{r-1},X_{t_{r-1}})\Delta t_r\), up to a remainder whose sum tends to zero in probability. To justify evaluating the time derivative at \(X_{t_{r-1}}\), rather than \(X_{t_r}\), use uniform continuity on the compact set reached by the stopped process: the accumulated difference is bounded by \(t\) times a modulus of continuity evaluated at \(\max_r|\Delta X_r|\), which tends to zero almost surely.

For the spatial change, Taylor's theorem at fixed time gives $$ f(t_{r-1},X_{t_r})-f(t_{r-1},X_{t_{r-1}}) =\sum_i f_{i,r-1}\Delta X_r^i+ \frac12\sum_{i,j}f_{ij,r-1}\Delta X_r^i\Delta X_r^j+R_r. $$ On a compact set, continuity of the Hessian bounds the remainder by \(|R_r|\le\omega(\max_q|\Delta X_q|)|\Delta X_r|^2\), where \(\omega(u)\to0\) as \(u\to0\). The stopped process has continuous paths, so \(\max_r|\Delta X_r|\to0\) almost surely. Also \(\sum_r|\Delta X_r|^2\) is bounded in probability: its coordinate sums converge to finite quadratic variations, and the cross terms are bounded by the corresponding sums of squares. Consequently \(\sum_r|R_r|\to0\) in probability. These arguments can be applied on successively larger stopping regions, giving the local result without assuming the process stays in a fixed compact set.

Summing the Taylor expansions leaves first-order sums and weighted quadratic sums. The drift parts of the first-order sums converge to ordinary time integrals; their Brownian parts converge to Ito integrals by the Ito isometry and approximation of continuous adapted integrands by their left-endpoint values. For the quadratic sums, use the standard quadratic-covariation theorem for Ito processes: for each \(i,j\), \(\sum_r\Delta X_r^i\Delta X_r^j\) converges in probability to \(\int_0^t c_{ij}(s)\,ds\). This is the quadratic-variation and covariation result for Ito processes; its partition-limit proof belongs to that earlier material. The weighted version applies to the continuous adapted weights \(f_{ij}(s,X_s)\), by approximating the weights with step processes and localizing. Thus the quadratic Taylor term converges to \(\frac12\int_0^t\sum_{i,j}f_{ij}(s,X_s)c_{ij}(s)\,ds\). The time, first-order, and quadratic limits are exactly the terms in the theorem, proving the formula. \(\square\)

Proof. The covariance matrix \(c\) and Hessian \(D^2f\) are symmetric. By the definition of matrix trace, \(\operatorname{tr}(cD^2f)=\sum_i\sum_j c_{ij}f_{ji}\). Equality of mixed partial derivatives gives \(f_{ji}=f_{ij}\), so this is precisely the double sum in the theorem. \(\square\)

Worked Applications

Why the Mixed Terms Matter

The differential multiplication rules \(dB_t^k\,dB_t^\ell=\mathbf{1}_{\{k=\ell\}}dt\), \(dB_t^k\,dt=0\), and \(dt\,dt=0\) summarize the relevant quadratic-covariation limits. In a vector process, the same Brownian coordinate can enter several coordinates of \(X\); their increments then have nonzero mixed covariation. The Hessian contraction records exactly these effects.

A common mistake is to retain only the diagonal terms. For the product example, that incorrect calculation would use \(f_{11}c_{11}=0\) and \(f_{22}c_{22}=0\), concluding that there is no correction. But \(f_{12}=f_{21}=1\) and \(c_{12}=c_{21}=1\), so the actual correction is \(\frac12(1+1)dt=dt\). The error amounts to pretending the coordinates have zero covariation even though both contain \(B^1\).

The Ito formula is also not the ordinary multivariable chain rule: its Hessian term comes from quadratic variation, with \([B]_t=t\), rather than from differentiability of Brownian paths. For Stratonovich integrals, a correction is needed when converting to Ito form; the formula proved here is for Ito integrals and uses the Ito correction directly. In particular, one must not remove the Hessian term merely by applying the ordinary chain rule.