ThmDex – An index of mathematical definitions, results, and conjectures.
Set of symbols
Deduction system
Zermelo-Fraenkel set theory
Binary cartesian set product
Binary relation
Simple map
Simple function
Measurable simple complex function
Simple integral
Unsigned basic integral
Unsigned basic expectation
Basic expectation
Random real number moment
Conditional expectation representative
Conditional expectation
Complex martingale
Definition D5698
Real martingale
Formulation 0
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) \begin{equation} \forall \, j \in J : \mathbb{E} |X_j| < \infty \end{equation}
(3) \begin{equation} \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) \end{equation}