ThmDex – An index of mathematical definitions, results, and conjectures.
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Definition D5698
Real martingale

Let $P = (\Omega, \mathcal{F}, \mathbb{P},\{ \mathcal{G}_j \}_{j \in J})$ be a D1726: Filtered probability space.
A D5697: Real random process $X : J \to \text{Random}(\Omega \to \mathbb{R})$ is a real martingale on $P$ if and only if
 (1) $X$ is a D2156: Adapted random collection on $P$ (2) $$\forall \, j \in J : \mathbb{E} |X_j| < \infty$$ (3) $$\forall \, i, j \in J \left( i < j \quad \implies \quad \mathbb{E}(X_j \mid \mathcal{G}_i) \overset{a.s.}{=} X_i \right)$$