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 = {∅, Ω, A, Ac} is the smallest σ-algebra Ω containing A.

      EXAMPLE 1.4.– If Ω is a topological space, the σ-algebra generated by the open sets of Ω is called the Borel σ-algebra of Ω. A Borel set is a set belonging to the Borel σ-algebra. On ℝ,

(ℝ) generally denotes the σ-algebra of Borel sets. It must be recalled that this is also the σ-algebra generated by the intervals, or by the intervals of the form ] − ∞, x], x ∈ ℝ. Thus, there is no unicity of the generating system.

      We will now recall the concept of the product σ-algebra.

      DEFINITION 1.4.– Let (Ei,

i)i∈ℕ be a sequence of measurable spaces.

       – Let n ∈ ℕ. The σ-algebra defined over and generated by

      is denoted by

0 ⊗ ... ⊗
n, and it is called the product σ-algebra over We have, in particular,

      In the specific case where E0 = ... = En = E and

0 = ... =
n = , we also write

       – We use ⊗i∈ℕi to denote the σ-algebra over the countable product space generated by the sets of the form where Ai ∈ i and Ai = Ei except for a finite number of indices i. In the specific case where, for any and i = , the product space is denoted by Eℕ, and the σ-algebra ⊗i∈ℕi is denoted by ⊗N.

      Finally, let us review the concepts of measurability and measure.

      DEFINITION 1.5.– Let Ω be non-empty set and

be a σ-algebra on Ω.

       – A measure over a probabilizable space (Ω, ) is defined as any mapping μ defined over , with values in [0, +∞] = ℝ+ ∪ {+∞}, such that μ(∅) = 0 and for any family (Ai)i∈ℕ of pairwise disjoint elements of , we have the property of σ-additivity:

       – A measure μ over a probabilizable space (Ω, ) is said to be finite, or have finite total mass, if μ(Ω) < ∞.

       – If μ is a measure over a probabilizable space (Ω, ), then the triplet (Ω, , μ) is called a measured space.

      DEFINITION 1.6.– Let (Ω,

) and (E, ε ) be two probabilizable spaces. A mapping X, defined over Ω taking values in E, is said to be (
, ε)-measurable, or just measurable, if there is no ambiguity regarding the reference σ-algebras, if

      In practice, when E ⊂ ℝ, we set ε =

(E) the set of Borel subsets of E, that is, the set of subsets of E. We can simply say that X is
-measurable. When, in addition, we manipulate a single σ-algebra
over Ω, it can be simply said that X is measurable. If we work with several σ-algebras over Ω, the concerned σ-algebra must always be specified: X is
-measurable.

      EXAMPLE 1.5.– If (Ω,

) is a measurable space and A ∈ , then the indicator function

      is

-measurable. Indeed, for any Borel set B in ℝ, we have

      Thus, in all cases, we do have

), (E, ε) and (G,
, ε) and (ε,
)-measurable mappings, respectively, then for any B ∈
, g−1(B) ∈ ε and consequently,

      Thus, the composition g ∘ f is indeed measurable on (Ω,

) in (G,
).

      We will now review the concept of a probability measure or probability distribution, and the concept of random variable, as well as the chief properties of these concepts.

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