Editing Martingale (probability theory) (section)

===Martingale sequences with respect to another sequence===

More generally, a sequence ''Y''<sub>1</sub>,&nbsp;''Y''<sub>2</sub>,&nbsp;''Y''<sub>3</sub>&nbsp;... is said to be a '''martingale with respect to''' another sequence ''X''<sub>1</sub>,&nbsp;''X''<sub>2</sub>,&nbsp;''X''<sub>3</sub>&nbsp;... if for all ''n''

:<math>\mathbf{E} ( \vert Y_n \vert )< \infty </math>

:<math>\mathbf{E} (Y_{n+1}\mid X_1,\ldots,X_n)=Y_n.</math>

Similarly, a '''[[continuous time|continuous-time]] martingale with respect to''' the [[stochastic process]] ''X<sub>t</sub>'' is a [[stochastic process]] ''Y<sub>t</sub>'' such that for all ''t''

:<math>\mathbf{E} ( \vert Y_t \vert )<\infty </math>

:<math>\mathbf{E} ( Y_{t} \mid \{ X_{\tau}, \tau \leq s \} ) = Y_s\quad \forall s \le t.</math>

This expresses the property that the conditional expectation of an observation at time ''t'', given all the observations up to time <math> s </math>, is equal to the observation at time ''s'' (of course, provided that ''s''&nbsp;≤&nbsp;''t''). The second property implies that <math>Y_n</math> is measurable with respect to <math>X_1 \dots X_n</math>.