Applications of martingale system theorems

Abstract

1. Basic definitions. The following definitions are all relative to a triple (Q, 13, Pr) in which Q is a space with points denoted by w, 13 is a Borel field of subsets of Q which includes Q, and Pr is a completely additive set function defined on sets of 13 such that for AC3B, 0<Pr {A} <Pr {Q} =1. A random variable is a function defined on Q having values on the real line extended by the adjunction of oo, + so and which is measurable with respect to 1B, i.e., for every real number r, the X set { x(X) ?r } is an element of 3. In the following the Borel fields discussed will be assumed to be subfields of 13 which include the set Q. If {xt, tCT} is any collection of random variables, the Borel field generated by this collection is the smallest Borel field with respect to which every member of the collection is measurable. A stochastic process is a collection {xt, Ft, tC T}, where T is a subset of the extended real line, { Ft, t CT } is an increasing collection of Borel fields in the sense that for t1 <t2, Ft1CFt2, and xt is a random variable measurable with respect to Ft or equal for almost all X to such a function. If the collection {Xt, tCT} is referred to as a stochastic process, it will be understood that Ft is the Borel field generated by {x8, s<t, sET} . By means of the probability measure Pr we define a Lebesgue integral and if a random variable x is integrable, we say that the expected value of x exists and define the expected value, written E{x}, by E{x} =fsxdpr. We do not require that E { x } be finite. Let x be a random variable such that E{x} exists and Fbe a Borel field. Then4 defined byk{A} =fAxdpr forAE F is a completely additive set function defined on F which is absolutely continuous with respect to probability measure, in the sense that C { A } = 0 if Pr {A } =0. By a generalization of the Radon-NikodYm theorem [5, p. 169] there exists a random variable y which is measurable with respect to F, for which E { y } exists, and such that 0 { A } =fAydpr. The random variable y is unique up to a set of probability measure zero. This discussion justifies the following definition. DEFINITION 1.1. Let x be a random variable such that E{x} exists and let F be a Borel field. The conditional expectation of x relative to F, written E{x||F}, is defined as any o function y which is equal for almost all X to a