Let \((X_{n}, \mathcal{F}_{n})\) be a martingale. The sequence of martingale differences \(\{ D_{n} \}_{n\geq 1}\) is defined by \(D_{n}=X_{n}-X_{n-1}\). Then \[ X_{n}=X_{0}+\sum_{i=1}^n D_{i}. \]
- Note: \(D_{n}\) is \(\mathcal{F}_{n}\) measurable, \(\mathbb{E}\lvert D_{n} \rvert < \infty\) and \(\mathbb{E}[D_{n+1}|\mathcal{F}_{n}]=0\) for all \(n\).