General Theory of Processes Notes

Sets subscripted with a “+” are nonnegative, and sets subscripted with a “++” are strictly positive. We write dxen := ∑ i∈N+ i+1 2n ·1[i/2n,(i+1)/2n)(x) and bxcn := ∑ i∈N+ i 2n ·1(i/2n,(i+1)/2n](x) so dxen x and bxcn x, unless x = 0, in which case bxcn = 0 for all n. We will write (Ft)0≤t≤∞ to denote a filtration. If X• is a process taking values in (E,E ), then F t = σ ({Xs ∈ A | 0 ≤ s ≤ t, A ∈ E }) is the natural filtration generated by X•. The following is nonstandard but convenient. Definition 0.1. We say that (Ft)0≤t≤∞ is close to (Gt)0≤t≤∞ if Gt− ⊆ Ft ⊂ Gt+ for each t. Proposition 0.2. If (Ft)0≤t≤∞ is close to (Gt)0≤t≤∞, then Ft+ = Gt+ and Ft− = Gt− for all t. Proof. Gt+1/3n ⊆ Ft+1/2n ⊆ Gt+1/n for each n, so the first result follows after intersecting over n. Similarly, Gt−1/n ⊆ Ft−1/2n ⊆ Gt−1/3n, for each n and the second result follows after taking the union and the σ-closure.