Theorem 1.2.4 (Integration by Parts). Let $(\Omega, \bracs{\cf_t|t \ge 0}, \bp)$ be a filtered probability space, $\bracs{X_t}$ be a $\bracs{\mathcal{F}_t}$-martingale, and $\phi: [0, \infty) \times \Omega \to \complex$ be a continuous, progressively measurable function. If:
For every $\omega \in \Omega$, $\phi(\cdot, \omega) \in BV_{\text{loc}}([0, \infty))$.
For all $t \ge 0$,
\[\ev\braks{\sup_{0 \le s \le t}|X_s|(|\phi|(s) + |\phi|(t))}< \infty\]
then the process
\[Y_{t} = X_{t} \phi(t) - \int_{0}^{t} X_{r} \phi(dr)\]
is a $\bracs{\mathcal{F}_t}$-martingale.