Martingales and Financial Mathematics in Discrete Time. Benoîte de Saporta

X, ifWe then write In the specific case where p = 2, we say there is a convergence in quadratic mean.

      2 2) The sequence of r.v. (Xn)n≥1 is called almost surely (a.s.) convergent towards X, if

       We then write

      THEOREM 1.1 (Monotone convergence theorem).– Let (X_n)_n≥1 be a sequence of positive and non-decreasing random variables and let X be an integrable random variable, all of these defined on the same probability space (Ω,

P). If (X_n) converges almost surely to X, then

      THEOREM 1.2 (Dominated convergence theorem).– Let (X_n)_n≥1 be a sequence of random variables and let X be another random variable, all defined on the same probability space (Ω,

, ℙ). If the sequence (X_n) converges to X a.s., and for any n ≥ 1, |X_n|≤ Z, where Z is an integrable random variable, then and, in particular,

      THEOREM 1.3 (Strong law of large numbers).– Let (X_n)_n≥1 be a sequence of integrable, independent random variables from the same distribution. Then,

1.3. Stochastic processes

      The main objective of this book is to study certain families of stochastic (or random) processes in discrete time. There are two ways of seeing such objects:

       – as a sequence (Xn)n∈ℕ of real random variables;

       – as a single random variable X taking values in the set of real sequences.

      The index n represents time. Since n ∈ ℕ, we speak of processes in discrete time. In the rest of this book, unless indicated otherwise, we will only consider processes taking discrete real values. The notation E thus denotes a finite or countable subset of ℝ and ε =

(E), the set of subsets of E.

      DEFINITION 1.18.– A stochastic process is a sequence X = (X_n)_n∈ℕ of random variables taking values in (E, ε). The process X is then a random variable taking values in (E^ℕ, ε^⊗ℕ).

      EXAMPLE 1.22.– A coin is tossed an infinite number of times. This experiment is modeled by Ω = {T, H}^ℕ∗ . For n ∈ ℕ^∗, consider the mappings X_n to Ω in ℝ defined by

      the number of tails at the nth toss. Therefore, X_n, n ∈ ℕ^∗ are discrete, real random variables and the sequence X = (X_n)_n∈ℕ is a stochastic process.

      DEFINITION 1.19.– Let X = (X_n)_n∈ℕ be a stochastic process. For all n ∈ ℕ, the distribution of the vector (X₀, X₁,..., X_n) is denoted by μ_n. The probability distributions (μ_n)_n∈ℕ are called finite-dimensional distributions or finite-dimensional marginal distributions of the process X = (X_n)_n∈ℕ.

      PROPOSITION 1.10.– Let X = (X_n)_n∈ℕ be a stochastic process and let (μ_n)_n∈ℕ be its finite-dimensional distributions. Then, for all n ∈ N∗ and (A₀,..., A_n−1) ∈ εⁿ, we have

      In other words, the restriction of the marginal distribution of the vector (X₀,..., X_n) to its first n coordinates is exactly the distribution of the vector (X₀,..., X_n−1).

      PROOF.– This proof directly follows from the definition of the objects. We have

      and hence, the desired equality.

      □

      Indeed, this property completely characterizes the distribution of the process X according to the following theorem.

      THEOREM 1.4 (Kolmogorov).– The canonical space (Ω,

) is defined in the following manner. Let Ω = E^ℕ. The coordinate mappings (X_n)_n∈ℕ are defined by X_n(ω) = ω_n for any ω = (ω_n)_n∈ℕ ∈ Ω and we write

= σ(X_n,n ∈ ℕ). Let (μ_n)_n∈ℕ be a family of probability distributions such that

      1 1) for any n ∈ ℕ, μn is defined on (En+1, ε⊗(n+1)),

      2 2) for any n ∈ ℕ∗ and (A0,..., An−1) ∈ εn, we have μn−1(A0 × ... × An−1) = μn(A0 × ... × An−1 × E).

      Therefore, there exists a unique probability distribution μ over the canonical space (Ω,

) such that the process X = (X_n)_n∈ℕ for the coordinate mapping has the distribution μ and for the finite-dimensional distributions has the sequence (μ_n)_n∈ℕ.

      This 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