Читать книгу Martingales and Financial Mathematics in Discrete Time онлайн
21 страница из 24
To study the random variables taking values in the set of sequences, we need new definitions for σ-algebras and measurability.
nn∈ℕ
– X is said to be adapted to the filtration (n)n∈ℕ (or again (n)n∈ℕ−adapted), if Xn is n-measurable for any n ∈ ℕ;
– X is said to be predictable with respect to the filtration (n)n∈ℕ (or again (n)n∈ℕ−predictable), if Xn is n−1-measurable for any n ∈ ℕ∗.
EXAMPLE 1.24.– A process is always adapted with respect to its natural filtration.
n
1.4. Exercises
EXERCISE 1.1.– Let Ω = {a, b, c}.
1 1) Completely describe all the σ-algebras of Ω.
2 2) State which are the sub-σ-algebras of which.
EXERCISE 1.2.– Let Ω = {a, b, c, d}. Among the following sets, which are σ-algebras?
1 1)
2 2)
3 3)
4 4)
For those which are not σ-algebras, completely describe the σ-algebras they generate.
– X1 be the random variable number of T on the first toss;
– X2 be the number of T on the second toss;
– Y be the number of T obtained on the two tosses;