Integration by parts for Brownian Ito processes

Normalize simultaneously as specified in the statement. The complement of the intersection of the two continuity/decomposition events and the countably many coefficient-integrability events belongs to ${\mathcal{F}}_0$ by [F1]. Multiplying all processes, initial values and coefficients by its full-event indicator preserves adaptation, progressive or predictable measurability as appropriate, all coefficient classes and the decompositions up to indistinguishability. The normalized $X,Y$ are continuous everywhere and therefore predictable. Any two such normalizations agree on a common full event, so their drift integrands agree pathwise there and their stochastic integrands agree in product measure on every finite horizon; the localized integral uniqueness preserves the resulting identities.

The differential integrals are well defined. Fix a finite $T$ and put $C_X={\sup }_{s\le T}\mathrm{\mid }{X}_s\mathrm{\mid }<\mathrm{\infty }$, $C_Y={\sup }_{s\le T}\mathrm{\mid }{Y}_s\mathrm{\mid }<\mathrm{\infty }$ pathwise. Then ${\int }_0^T\mathrm{\mid }{X}_s{b}_s^Y\mathrm{\mid }ds\le C_X{\int }_0^T\mathrm{\mid }{b}_s^Y\mathrm{\mid }ds<\mathrm{\infty }$ and ${\int }_0^T(X_s{\eta }_s{)}^{2}ds\le C_X^2{\int }_0^T{\eta }_s^{2}ds<\mathrm{\infty }$; the corresponding two bounds with $X,Y$ exchanged also hold. The products in the drift are progressive, and those in the stochastic terms are predictable. Moreover $2\mathrm{\mid }\xi \eta \mathrm{\mid }\le {\xi }^2+{\eta }^2$ makes the covariance integral locally finite and continuous. For $U=X+Y$, the combined drift is locally absolutely integrable and its diffusion satisfies $(\xi +\eta {)}^2\le 2{\xi }^2+2{\eta }^2$; thus $U$ is a class process with coefficients $b^X+b^Y,\xi +\eta$. The same bounds ensure the integrability of all products used below, including $X\xi ,Y\eta$ and $U(\xi +\eta )$.

Apply [F3] with $f(t,z)=z^2$, whose time derivative is zero, first space derivative is $2z$ and second space derivative is $2$, separately to $X,Y,U$. Each application is licensed by step 2.1 and gives, on one common full event for all $t$, $X_t^2-X_0^2=2{\int }_0^t{X}_s{b}_s^{X}ds+2{\int }_0^t{X}_s{\xi }_{s}d{B}_s+{\int }_0^t{\xi }_s^{2}ds,$ $Y_t^2-Y_0^2=2{\int }_0^t{Y}_s{b}_s^{Y}ds+2{\int }_0^t{Y}_s{\eta }_{s}d{B}_s+{\int }_0^t{\eta }_s^{2}ds,$ $U_t^2-U_0^2=2{\int }_0^t{U}_s(b_s^X+b_s^Y)ds+2{\int }_0^t{U}_s({\xi }_s+{\eta }_s)d{B}_s+{\int }_0^t({\xi }_s+{\eta }_s{)}^{2}ds\mathrm{.}$

Subtract the first two identities from the third and divide by two. The left side is $X_t{Y}_t-X_0{Y}_0$. The drift coefficient is $X{b}^Y+Y{b}^X$, the stochastic coefficient is $X\eta +Y\xi$, and the last coefficient is $\xi \eta$. Pathwise Lebesgue linearity applies by the absolute-integrability bounds in step 2.1. To justify stochastic subtraction and separation, stop at the common energy levels of the finitely many product integrands in step 2.1, also capped by the integer level in time. Each stopped integral has finite energy, so finite-energy linearity holds there. The stopping identity and countable exhaustion give the same linearity up to indistinguishability for the original localized integrals. Consequently $X_t{Y}_t-X_0{Y}_0={\int }_0^t(X_s{b}_s^Y+Y_s{b}_s^X)ds+{\int }_0^t{X}_s{\eta }_{s}d{B}_s+{\int }_0^t{Y}_s{\xi }_{s}d{B}_s+{\int }_0^t{\xi }_s{\eta }_{s}ds$ on a common full event for all $t$.

By [F4] the last integral in step 4.1 equals $[X,Y]$ up to indistinguishability, and step 2.1 identifies the first three terms as the two differential integrals in the statement. Intersect the finitely many full events. This proves the stated identity for the normalized versions at all times, and step 1.1 transfers it to the original processes.

Suppose now that $b^X=b^Y=0$. By step 5.1 the centered process $XY-[X,Y]-X_0{Y}_0$ is the sum of two localized Ito integrals. Common energy localization of $X\eta ,Y\xi$ makes this a continuous local martingale by [F2]. If $X_0{Y}_0$ is integrable, the ${\mathcal{F}}_0$-measurable constant process with that value is a martingale, so [F2] also makes $XY-[X,Y]$ a local martingale. Under the additional expectation hypotheses at a fixed $t$, the two stochastic integrals are individually square-integrable and mean zero, and $E\mathrm{\mid }[X,Y{]}_t\mathrm{\mid }\le E{\int }_0^t\mathrm{\mid }{\xi }_s{\eta }_s\mathrm{\mid }ds<\mathrm{\infty }$. Together with integrability of $X_0{Y}_0$, the product identity proves integrability of $X_t{Y}_t$ itself. Taking expectations yields exactly the stated formula. For nonzero drifts the drift integrals remain and no local-martingale corollary is asserted.

At $t=0$ every integral is zero. A constant factor has zero diffusion and gives its constant-multiple product identity. If either diffusion vanishes, the covariation vanishes while any remaining drift and stochastic terms stay in the displayed formula; if both vanish this is the ordinary finite-variation product rule. For $X=Y$ the identity becomes $X_t^2=X_0^2+2{\int }_0^t{X}_{s}d{X}_s+[X{]}_t$, already obtained in step 3.1. No boundedness of the original initial values or of the diffusion coefficients was assumed. All energy bounds are local pathwise bounds converted to expected bounds only by stopping, and full AC is inherited as in [F5]. These cases are consistent with the zero-drift restriction in step 6.1.