 
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
subalgebras of F. A sequence of random variables (Xt) is adapted to a filtration (Ft) if Xt is
Ftmeasurable 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[XA]] = EX (the socalled "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 subalgebra, F1 F2, and have E [X  F2] = Y for some variable
Y F1, then
