Martingale transform. Let \(\left(u_{n}, \mathscr{A}_{n}ight)_{n \in \mathbb{N}}\) be a martingale and let \(\left(f_{n}ight)_{n \in \mathbb{N}}\) be a sequence of bounded functions such that \(f_{n} \in \mathcal{M}\left(\mathscr{A}_{n}ight)\) for every \(n \in \mathbb{N}\). Set \(f_{0}:=0\) and \(u_{0}:=0\). Then the so-called martingale transform

\[(f \cdot u)_{n}:=\sum_{i=1}^{n} f_{i-1} \cdot\left(u_{i}-u_{i-1}ight), \quad n \in \mathbb{N}\]

is again a martingale w.r.t. \(\left(\mathscr{A}_{n}ight)_{n \in \mathbb{N}}\).