University of Washington Math 523A Lecture 1 Martingales: definitions and examples Summary: University of Washington Math 523A Lecture 1 Martingales: definitions and examples Lecturer: Eyal Lubetzky March 30, 2009 Basic definitions Let (, F) be a measurable space. A filtration F0 F1 F2 . . . F is an increasing sequences of sub--algebras of F. A sequence of random variables (Xt) is adapted to a filtration (Ft) if Xt is Ft-measurable for all t. Given a stochastic process, one can think of (Ft) as the "history so far". In many cases, it will be useful to consider the natural filtration generated by Xt, i.e., Ft = (X0, . . . , Xt) is the smallest -algebra in which {Xi : i t} are measurable. In other cases, the stochastic process will include some extra randomness beyond that which is observed in the variables Xi. Throughout the course, we will use the fact that E[E[X|A]] = EX (the so-called "tower of expectations" property). In particular, if we are conditioning on a sequence of random variables and have E [X | Y1, Y2, . . . , Yk] = f(Y1) for some function f, then E [X | Y1] = E [E [X | Y1, Y2, . . . , Yk] | Y1] = E [f(Y1) | Y1] = f(Y1) . In general, whenever we have a sub--algebra, F1 F2, and have E [X | F2] = Y for some variable Y F1, then Collections: Computer Technologies and Information Sciences