Stochastic Integration by Parts

The integration by parts formula for stochastic processes generalizes the classical product rule, incorporating the quadratic covariation term.

Formula

Theorem6.1Integration by parts

For Itô processes $X_t$ and $Y_t$ :

$d(X_t Y_t) = X_t \, dY_t + Y_t \, dX_t + dX_t \cdot dY_t,$

where the last term is the quadratic covariation $d\langle X, Y \rangle_t$ .

In differential notation: if $dX_t = \mu_X dt + \sigma_X dB_t$ and $dY_t = \mu_Y dt + \sigma_Y dB_t$ , then $dX_t \cdot dY_t = \sigma_X \sigma_Y dt$ .

Example$B_t \cdot B_t$

$d(B_t^2) = B_t \, dB_t + B_t \, dB_t + dB_t \cdot dB_t = 2B_t \, dB_t + dt.$

This matches Itô's formula with $f(x) = x^2$ .

Application: Variance of integral

ExampleVariance of $\int_0^t B_s \, dB_s$

Let $I_t = \int_0^t B_s \, dB_s$ . By Itô, $I_t = \frac{1}{2}B_t^2 - \frac{1}{2}t$ . Using integration by parts:

$\text{Var}(I_t) = \text{Var}(B_t^2/2 - t/2) = \frac{1}{4}\text{Var}(B_t^2) = \frac{1}{4} \cdot 2t^2 = \frac{t^2}{2}.$

Summary

Integration by parts: $d(XY) = X \, dY + Y \, dX + dX \cdot dY$ , with the covariation term essential in stochastic calculus.