the multidimensional Ito formula
/ EE-toh /
The multidimensional Ito formula extends the change-of-variables rule to a function of several semimartingales driven by several Brownian motions. It is exactly what you need to do real work: most models live in several dimensions (a portfolio of assets, a system of coupled SDEs, the position of a particle in space), and the second-order corrections now involve a full matrix of cross terms, not just a single (1/2) f''. The new ingredient compared with the scalar case is the quadratic COvariation between components.
Let X = (X^1, ..., X^d) be a vector of continuous semimartingales, and let f : R^d -> R be twice continuously differentiable. Then f(X_t) = f(X_0) + sum_i integral_0^t partial_i f(X_s) dX^i_s + (1/2) sum_{i,j} integral_0^t partial_i partial_j f(X_s) d[X^i, X^j]_s, where partial_i f is the i-th partial derivative, partial_i partial_j f is the Hessian entry, and [X^i, X^j] is the quadratic covariation of components i and j. In SDE form, if dX^i = b^i dt + sum_a sigma^{i a} dB^a with B^1, ..., B^m independent Brownian motions, then d[X^i, X^j] = (sum_a sigma^{i a} sigma^{j a}) dt = (sigma sigma^T)^{i j} dt, so the correction term is (1/2) sum_{i,j} partial_i partial_j f * (sigma sigma^T)^{i j} dt, i.e. it is the trace of the Hessian against the diffusion matrix a = sigma sigma^T. The shorthand multiplication table is dB^a dB^b = delta_{ab} dt (independent Brownians have zero covariation, a Brownian has [B^a] = t) and dB^a dt = (dt)^2 = 0. The first-order sum is the ordinary gradient-times-increment, exactly as in classical multivariate calculus; the matrix second-order sum is what is new.
This is the workhorse for diffusions. Writing the correction as (1/2) trace(a D^2 f) shows immediately that the second-order operator L f = sum b^i partial_i f + (1/2) sum a^{i j} partial_i partial_j f is the generator of the diffusion: f(X_t) - integral_0^t L f(X_s) ds is a local martingale, which is the multidimensional Dynkin / Ito-to-generator link and the route to Feynman-Kac and to the connection with elliptic and parabolic PDEs. The honest caveats carry over and add one: f must be C^2; the formula is for the Ito convention (Stratonovich obeys the ordinary chain rule); and crucially the cross terms d[X^i, X^j] are NOT generally zero — only if the driving Brownian motions are independent AND the diffusion matrix is diagonal do the off-diagonal corrections vanish. Forgetting the i != j covariation terms is the most common mistake in multidimensional Ito calculations.
Let B^1, B^2 be independent Brownian motions and f(x, y) = x y. The Hessian has partial_x partial_y f = 1 and zero diagonal, and [B^1, B^2] = 0 because the Brownians are independent. Ito gives d(B^1 B^2) = B^2 dB^1 + B^1 dB^2 + d[B^1, B^2] = B^2 dB^1 + B^1 dB^2 (no correction). But if instead the two coordinates were the SAME Brownian, f(B, B) = B^2 picks up the diagonal term d[B] = dt, recovering d(B^2) = 2B dB + dt — the covariation, not independence, is what decides the correction.
Independent components have zero covariation (no cross correction); a shared driver brings back the dt term.
The off-diagonal corrections (1/2) partial_i partial_j f d[X^i, X^j] vanish only when components have zero quadratic covariation (independent drivers, diagonal diffusion); never assume the correction is just a sum of single-variable (1/2) f'' d[X^i] terms.