Concept guide
Finite filtration: checking information nesting by partition refinement, and separating it from adaptedness and independence
On Ω={0,1,2,3} (two coin tosses: 0=HH, 1=HT, 2=TH, 3=TT), represent the σ-algebras at times 0, 1, 2 by partition rows P_0, P_1, P_2. Prove that 'the later partition refines the earlier one ⇔ the σ-algebras are nested', check every adjacent pair of times completely and conclude that (F_t) is a filtration; give a sequence of partitions that is not a filtration; and show that a filtration neither makes every process adapted nor implies independence.