Martingales and Financial Mathematics in Discrete Time. Benoîte de Saporta

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      1

      Elementary Probabilities and an Introduction to Stochastic Processes

      This chapter reviews the basic concepts related to probability and random variables which will be useful for the rest of this text. For a more detailed explanation as well as demonstrations, the readers may refer to [BAR 07, DAC 82, FOA 03, OUV 08, OUV 09] in French and [BIL 12, CHU 01, DUR 10, KAL 02, SHI 00] in English. The readers who are already familiar with these concepts may proceed straight to section 1.3, which introduces the concept of stochastic processes.

      This chapter begins with a brief summary of the concepts of a σ-algebra in section 1.1. These concepts will help in understanding the construction of the properties of conditional expectation in Chapter 2. We then study the chief definitions and properties of random variables and their distribution in section 1.2. There is an emphasis on discrete random variables as this entire book essentially studies discrete cases. Section 1.3 defines a stochastic process, which is the main subject studied in this book. Finally, there are exercises in handling these different concepts in section 1.4. The solutions are given in Chapter 8.

      Throughout the rest of the text, Ω is a non-empty set and

(Ω) denotes the set of the subsets of Ω :

      The set Ω is called the universe or the fundamental set. In practice, the set Ω contains all the possible outcomes of a random experiment.

1.1. Measures and σ-algebras

      Let us start by reviewing the concept of a σ-algebra.

      DEFINITION 1.1.– A subset

(Ω) is a σ-algebra over Ω if

      1 1) Ω ∈ ;

      2 2) is stable by complementarity: for any A ∈ , we have Ac ∈ , where Ac denotes the complement of A in Ω: Ac = Ω\A;

      3 3) is stable under a countable union: for any sequence of elements (An)n∈ℕ of , we have

      Elements of a σ-algebra are called events.

      EXAMPLE 1.1.– The set

, we clearly have

.

      EXAMPLE 1.2.– The set

(Ω) is the largest σ-algebra over Ω; it is called the largest σ-algebra. Indeed, by construction,

(Ω) contains all the subsets of Ω, and thus it contains in particular Ω and it is stable by complementarity and under countable unions. In addition, any other σ-algebra

(Ω).

      DEFINITION 1.2.– Let Ω be a non-empty set and

be a σ-algebra over Ω. The couple (Ω,

) is called a probabilizable space.

      Among the elementary properties of σ-algebra, we can cite stability through any intersection (countable or not).

      PROPOSITION 1.1.– Any intersection of σ-algebras over a set Ω is a σ-algebra over Ω.

      PROOF.– Let (

_i)_i∈I be any family of σ-algebra indexed by a non-empty set I. Thus,

       – first of all, for any i, Ω ∈ i, thus Ω ∈ ∩i∈Ii;

       – secondly, if A ∈ ∩i∈I i, then for any i, A ∈ i. As these are σ-algebras, we have that for any i, Ac ∈ i, thus Ac ∈ ∩i∈I i;

       – finally, if for any n ∈ ℕ, An ∈ ∩i∈I i, then for any i, n, An ∈ i. As these are σ-algebras, we have that for any i, ∪n∈ℕAn ∈ i, thus

      It is generally difficult to make explicit all the events in a σ-algebra. We often describe it using generating events.

      DEFINITION 1.3.– Let ε be a subset of

(Ω). The σ-algebra σ(ε) generated by ε is the intersection of all σ-algebras containing ε. It is the smallest σ-algebra containing ε. ε is called the generating system of the σ-algebra σ(ε).

      It can be seen that σ(ε) is indeed a σ-algebra, being an intersection

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