x.(u1) x.( Figure 3: Sample paths of stochastic process X If Xnlnso is real valued, then Nmo can be viewed as a random variable with values in R(path space; here N is the set of integers) It is important to quantify the information history or flow corresponding to a stochastic rocess. This is achieved using filtration Definition 3.5 A filtration of a measurable space( @, F)is a family IFninso of sub algebra n C such that Fn Cn+1, Vn=0, 1 Example 3.6 If (Xn) is a stochastic process 9n=a(X0,X0,…,Xn) defines a filtration 9n, gn C F called the history of Xn) or filtration generated by INh Definition 3.7 A stochastic process (Xn) is adapted to a filtration (Fn if Xn is Fn measurable for all n=0, 1, 2 Remark 3.8 We can consider Fn are events that have occurred up to timen ✻ ✑ ✑ ✑ ✑ ✑✑❅ ❅ ❅ ❅ ❅❅☞ ☞ ☞ ☞ ☞ ☞ ☞ ☞ ☞ ☞ ☞ ☞ ☞ ☞❅❅ ❅ ❅ ❅❅✚ ✚ ✚ ✚ ✚✚ ❩❩❩❩❩❩✪ ✪ ✪ ✪ ✪ ✪ ❍❍❍❍❍❍❍✏✏✏✏✏✏❩❩❩❩❩❩ 1 2 3 4 5 X·(ω1) X·(ω2) ✲ Figure 3: Sample paths of stochastic process X. If {Xn} ∞ n=0 is real valued, then {Xn} ∞ n=0 can be viewed as a random variable with values in RN (path space; here N is the set of integers). It is important to quantify the information history or flow corresponding to a stochastic process. This is achieved using filtration. Definition 3.5 A filtration of a measurable space (Ω, F) is a family {Fn} ∞ n=0 of sub-σ- algebra Fn ⊂ F such that Fn ⊂ Fn+1 , ∀ n = 0, 1, . . . Example 3.6 If {Xn} is a stochastic process, Gn 4 = σ(X0, X0, . . . , Xn) defines a filtration {Gn}, Gn ⊂ F called the history of {Xn} or filtration generated by {Xn}. Definition 3.7 A stochastic process {Xn} is adapted to a filtration {Fn} if Xn is Fn￾measurable for all n = 0, 1, 2, . . .. Remark 3.8 We can consider Fn are events that have occurred up to time n. 26