Martingales and Financial Mathematics in Discrete Time. Benoîte de Saporta

inclusion) of sub-σ-algebras of

.

      When (

_n)_n∈ℕ is a filtration defined on the probability space (Ω,

, ℙ), the quadruplet (Ω,

, ℙ, (

_n)_n∈ℕ) is said to be a filtered probability space.

      EXAMPLE 1.23.– Let (X_n)_n∈ℕ be a sequence of random variables and we consider, for any n ∈ ℕ,

_n = σ(X₀, X₁, ..., X_n), the σ-algebra generated by {X₀, ..., X_n}. The sequence (

_n)_n∈ℕ is, therefore, a filtration, called a natural filtration of (X_n)_n∈ℕ or filtration generated by (X_n)_n∈ℕ. This filtration represents the information revealed over time, by the observation of the drawings of the sequence X = (X_n)_n∈ℕ.

      DEFINITION 1.21.– Let (Ω,

, ℙ, (

_n)_n∈ℕ) be a filtered probability space, and let X = (X_n)_n∈ℕ be a stochastic process.

       – 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.

      As its name indicates, for a predictable process, we know its value X_n from the instant n − 1.

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.

      EXERCISE 1.3.– Let X be a random variable on (Ω,

) and let

. Show that X is

-measurable if and only if σ(X) ⊂

.

      EXERCISE 1.4.– Let A ∈

and let

. Show that

-measurable if and only if A ∈

.

      EXERCISE 1.5.– Let Ω = {P, F} × {P, F} and =

=

(Ω), corresponding to two successive coin tosses. Let

       – 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;

       – and Z = 1 if the two tosses yielded an identical result; otherwise, it is 0.

      1 1) Describe 1 = σ(X1) and 2 = σ(X2). Is X1 2-measurable?

      2 2) Describe = σ(Y). Is Y 1-measurable? Is X1 -measurable?

      3 3) Describe = σ(Z). Is Z 1-measurable, -measurable? Is X1 -measurable?

      4 4) Give the inclusions between , 1, 2, and .

      EXERCISE 1.6.– Let (X_n)_n≥1 be a sequence of independent random variables with the same Rademacher distribution with parameter 1/2:

      Let S₀ = 0 and The process S = (S_n)_n∈ℕ is called the simple symmetric random walk on . It will be studied in detail in Chapter 3. We write X₀ = 0. Show that the filtration generated by the sequence (X_n)_n∈ℕ is the same as that generated by the sequence (S_n)_n∈ℕ.

      EXERCISE 1.7.– Consider the following game of chance. A player begins by choosing a number between 6 and 8 (inclusive), which we call the principal. The player then rolls 2 uncut, six-sided, non-rigged dice and sums the result. The wins are as follows:

       – If the sum is 2 or 3, the player loses 1 DT (Tunisian dinar).

       – If the sum is 11, the player wins 1 DT if the principal is 7; otherwise, they lose 1 DT.

       – If the sum is 12, the player wins 1 DT if the principal is 6 or 8; otherwise, they lose 1 DT.

       – Finally, in all other cases, nothing happens (no win, no loss).

      1 1) Determine Ω, the universe of all outcomes of the experiment.

      2 2) S is the random variable giving the sum of the two dice. Determine the distribution of S.

      3 3) X7 is the random variable giving