the stochastic integration-by-parts formula
/ EE-toh /
The stochastic integration-by-parts (or product) formula tells you how to differentiate a product of two semimartingales, and it is where the quadratic covariation makes its first explicit appearance. In ordinary calculus the Leibniz rule says d(XY) = X dY + Y dX. In stochastic calculus there is an extra term: d(XY) = X dY + Y dX + d[X, Y], where [X, Y] is the quadratic covariation. It is really just the multidimensional Ito formula applied to f(x, y) = xy, but it is used so constantly that it earns its own name and is the cleanest way to introduce covariation.
Precisely, for continuous semimartingales X and Y, the product XY is again a semimartingale and X_t Y_t = X_0 Y_0 + integral_0^t X_s dY_s + integral_0^t Y_s dX_s + [X, Y]_t. The covariation [X, Y]_t is the limit, as the partition mesh goes to 0, of sum (X_{t_{k+1}} - X_{t_k})(Y_{t_{k+1}} - Y_{t_k}) — the sum of products of increments — and it is the symmetric bilinear companion of the quadratic variation, with the polarization identity [X, Y] = ([X+Y] - [X] - [Y]) / 2. Equivalently in differential shorthand the rule says (X + dX)(Y + dY) - XY = X dY + Y dX + dX dY, and dX dY is read off the Ito multiplication table; for X driven by dB and Y by dW with d[B, W] = rho dt one gets dX dY = sigma_X sigma_Y rho dt. Covariation is symmetric and bilinear, vanishes if either factor has bounded variation (since then no second-order term survives), and for two independent Brownian motions is zero.
It is the everyday tool for computing with products: deriving the dynamics of XY, integrating the stochastic exponential, computing covariances of stochastic integrals, and proving theorems like Levy's characterization and the martingale representation. The honest caveats: the correction d[X, Y] is present whenever BOTH factors have a martingale (or unbounded-variation) part with nonzero covariation; if one factor is of bounded variation (e.g. a deterministic time integral), [X, Y] = 0 and you recover the ordinary Leibniz rule. Do not forget the bracket term — omitting it is the integration-by-parts version of forgetting the (1/2) f'' in Ito's formula. And [X, Y] is a covariation of the integrators, computed from the multiplication table, not from time alone.
Let X_t = B_t and Y_t = B_t (the same Brownian motion). Then [X, Y] = [B] = t, and the product rule gives d(B B) = B dB + B dB + d[B] = 2B dB + dt, recovering d(B^2) = 2B dB + dt. For two independent Brownian motions B and W, [B, W] = 0, so d(BW) = B dW + W dB with no correction — illustrating that the bracket term is exactly the covariation, present for correlated drivers and absent for independent ones.
Same driver: the +dt bracket appears. Independent drivers: zero covariation, ordinary Leibniz.
The covariation term [X, Y] is zero if either factor has bounded variation, so integration by parts against a deterministic or finite-variation integrand looks ordinary; the correction only appears between two genuinely rough (martingale) parts.