2. For each t E Z+ and each Borel set B CB(R P(Xt+1∈BFt)=P(Xt+1∈B|a(Xt) ometimes when the probability measure and filtration are understood, we will talk simply of a Markov process Remark. Often the filtration F is taken to be that generated by the process X itself Proposition. Let(Q, F, P, E)be a filtered probability space and let X be a(P, F)- Markov process. Let t, k E Z+. Let f: R-R be a Borel function such that f(Xt+k) is integrable. Then EIf(Xt+)Ft]=EI(Xi+)o(Xi) 10 and hence there is a borel function g:Z+×Z+×R→ R such that, for each t, EI(Xu+k)Ft=g(t, k, Xt) 11 Proof. To show it for k:= 1, use that f(Xi+1) is a o(X++1)-measurable random variable and hence is the limit of a monotone increasing sequence of o(Xt+1)-simple random variables. But such random variables are linear combinations of indicator functions of sets X+1(B)with B a Borel set. This completes the proof for k=1 To prove it for arbitrary positive k, use induction. To prove it for k+ 1 assuming it true for k use the law of iterated The vector case is a simple extension of the scalar case. However, it is important that the definition of a vector Markov process is not that each component is Markov2. For each t ∈ Z+ and each Borel set B ⊂ B (R) P (Xt+1 ∈ B|Ft) = P (Xt+1 ∈ B|σ (Xt)). (9) Sometimes when the probability measure and filtration are understood, we will talk simply of a Markov process. Remark. Often the filtration F is taken to be that generated by the process X itself. Proposition. Let (Ω, F, P, F) be a filtered probability space and let X be a (P, F)- Markov process. Let t, k ∈ Z+. Let f : R → R be a Borel function such that f (Xt+k) is integrable. Then, E [f (Xt+k)|Ft ] = E [f (Xt+k)|σ (Xt)] (10) and hence there is a Borel function g : Z+ × Z+ × R → R such that, for each t, E [f (Xt+k)|Ft ] = g (t, k, Xt). (11) Proof. To show it for k = 1, use that f (Xt+1) is a σ (Xt+1) −measurable random variable and hence is the limit of a monotone increasing sequence of σ (Xt+1) −simple random variables. But such random variables are linear combinations of indicator functions of sets X −1 t+1 (B) with B a Borel set. This completes the proof for k = 1. To prove it for arbitrary positive k, use induction. To prove it for k + 1 assuming it true for k, use the law of iterated expectations. The vector case is a simple extension of the scalar case. However, it is important that the definition of a vector Markov process is not that each component is Markov. 12