Martingales and Financial Mathematics in Discrete Time. Benoîte de Saporta
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Название: Martingales and Financial Mathematics in Discrete Time

Автор: Benoîte de Saporta

Издательство: John Wiley & Sons Limited

Жанр: Математика

Серия:

isbn: 9781119885023

isbn:

СКАЧАТЬ result is very important for the theory of processes as it signifies that it is sufficient to specify (all) the finite-dimensional distributions and for them to be compatible with each other, to uniquely define a process distribution over the space of infinite random sequences. In practice, this makes it possible to justify the construction of processes (existence property) as well as showing that two processes have the same distribution (unicity property).

      To study the random variables taking values in the set of sequences, we need new definitions for σ-algebras and measurability.

      DEFINITION 1.20.– In a probability space (Ω,

, ℙ), a filtration is a sequence (
n)n∈ℕ of sub-σ-algebras of
such that, for any n ∈ ℕ,
n ⊂
n+1. This is, thus, a non-decreasing sequence (for 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 (Xn)n∈ℕ be a sequence of random variables and we consider, for any n ∈ ℕ,

n = σ(X0, X1, ..., Xn), the σ-algebra generated by {X0, ..., Xn}. The sequence (
n)n∈ℕ is, therefore, a filtration, called a natural filtration of (Xn)n∈ℕ or filtration generated by (Xn)n∈ℕ. This filtration represents the information revealed over time, by the observation of the drawings of the sequence X = (Xn)n∈ℕ.

      DEFINITION 1.21.– Let (Ω,

, ℙ, (
n)n∈ℕ) be a filtered probability space, and let X = (Xn)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 Xn from the instant n − 1.

      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 СКАЧАТЬ