2.2 Filtrations Let(Q, F)be a measureable space. A filtration in discrete time is a sequence of r-algebras Ft) such that FtCy Ft Ft+ for all t=0.1... In continuous time, the second condition is replaced by Fs CFt for all s< t 3 Markov processes The idea of a Markov process is to capture the idea of a short-memory stochastic process: once its current state is known, past history is irrelevant from the point of view of predicting its future Definition. Let( Q, F)be a measurable space and let(P, E)be, respectively,a probability measure on and a filtration of this space. Let X be a stochastic process in discrete time on( Q, F). Then X is called a(P, E )-Markov process if 1. X is F-adapted, and2.2 Filtrations Let (Ω, F) be a measureable space. A filtration in discrete time is a sequence of σ–algebras {Ft} such that Ft ⊂ F and Ft ⊂ Ft+1 for all t = 0, 1, . . .. In continuous time, the second condition is replaced by Fs ⊂ Ft for all s ≤ t. 3 Markov processes The idea of a Markov process is to capture the idea of a short-memory stochastic process: once its current state is known, past history is irrelevant from the point of view of predicting its future. Definition. Let (Ω, F) be a measurable space and let (P, F) be, respectively, a probability measure on and a filtration of this space. Let X be a stochastic process in discrete time on (Ω, F). Then X is called a (P, F)-Markov process if 1. X is F−adapted, and 11