Instead, we require that all the relevant(for the future of X) bits of information in Ft are in the a-algebra generated by all the stochastic variables in Xt, i.e. o(Xt)is defined as the single a-algebra o(Xi.t, X2,t,,Xn,t). This means that, for each Borel function f:Rn→R EI(Xi+k)]= EI(X++k)lo(Xt) and hence that there is a Borel function g: Z+ X R"-R such that, for each t EIf(X+1ft=g(t, Xt (13 (The case k= l is so important that we stress it here by ignoring greater values of 3.0.1 Probability transition functions and time homogeneity Definition. Let(, F, P, E be a filtered probability space and let X be a(p F Markov process. Then, for each t=0, 1, 2... its probability transition function Qt:R×B(R)→[0.,1 is defined via Qt(x,B)=P(Xt∈BXt Note that any Markov process has a sequence of probability transition functions Note also that for each fixed t and a, Qt+1(, )is a probability measure on B(R) Meanwhile, if we fix B, Q++1(X(), B)is a random variable. Indeed, it is the con ditional probability of Xt+1 E B given Xt, i.e. Q++1(Xt, B)=EIx41(o(X. Moreover, the conditional expectation of any a(Xt+1)-measurable random variable (given X, is an integral with respect to the measure Qt+1 in the following senseInstead, we require that all the relevant (for the future of X) bits of information in Ft are in the σ−algebra generated by all the stochastic variables in Xt , i.e. σ (Xt) is defined as the single σ−algebra σ ({X1,t, X2,t, ..., Xn,t}). This means that, for each Borel function f : R n → R m, E [f (Xt+k)|Ft ] = E [f (Xt+k)|σ (Xt)] (12) and hence that there is a Borel function g : Z+ × R n→ R m such that, for each t, E [f (Xt+1)|Ft ] = g (t, Xt). (13) (The case k = 1 is so important that we stress it here by ignoring greater values of k.) 3.0.1 Probability transition functions and time homogeneity Definition. Let (Ω, F, P, F) be a filtered probability space and let X be a (P, F)- Markov process. Then, for each t = 0, 1, 2... its probability transition function Qt : R×B (R) → [0, 1] is defined via Qt (x, B) = P (Xt ∈ B|Xt−1 = x). (14) Note that any Markov process has a sequence of probability transition functions. Note also that for each fixed t and x, Qt+1 (x, ·) is a probability measure on B (R). Meanwhile, if we fix B, Qt+1 (Xt (·), B) is a random variable. Indeed, it is the con￾ditional probability of Xt+1 ∈ B given Xt , i.e. Qt+1 (Xt , B) = E h IX −1 t+1(B) ¯ ¯ ¯ σ (Xt) i . Moreover, the conditional expectation of any σ (Xt+1)-measurable random variable (given Xt) is an integral with respect to the measure Qt+1 in the following sense. 13