5: FILTRATIONS AND CONDITIONING Marek Rutkowski School of Mathematics and Statistics University of Sydney Semester 2, 2016 M. Rutkowski (USydney) Slides 5: Filtrations and Conditioning 1 / 37 New Features In the case of multi-period market models, we need to deal with two important new features. First, the investors can trade in assets at any specific date t ∈ {0, 1, 2,...,T } where T is the horizon date. Second, the investors can gather information over time, since the fluctuations of asset prices can be observed. We will need to introduce the concept of a self-financing trading strategy. We have to determine how the level of information available to investors evolves over time. The latter aspect leads to the probabilistic concepts of σ-fields and filtrations. M. Rutkowski (USydney) Slides 5: Filtrations and Conditioning 2 / 37 Outline We will examine the following issues: 1 Partitions and σ-fields. 2 Filtrations and adapted stochastic processes. 3 Conditional expectation with respect to a partition or a σ-field. 4 Change of a probability measure and the Radon-Nikodym density. M. Rutkowski (USydney) Slides 5: Filtrations and Conditioning 3 / 37 PART 1 PARTITIONS AND σ-FIELDS M. Rutkowski (USydney) Slides 5: Filtrations and Conditioning 4 / 37 σ-Field The concept of a σ-field can be used to describe the amount of information available at a given moment. Let N = {1, 2,... } be the set of all natural numbers. Definition (σ-Field) A collection F of subsets of Ω is called a σ-field (or a σ-algebra) whenever: 1 Ω ∈ F, c 2 if A ∈ F then A := Ω \ A ∈ F, ∞ 3 if Ai ∈ F for all i ∈ N then i=1 Ai ∈ F. M. Rutkowski (USydney) Slides 5: Filtrations and Conditioning 5 / 37 Interpretation of a σ-Field The set of information has to contain all possible states, so that we postulate that Ω belongs to any σ-field. Any set A ∈ F is interpreted as an observed event. If an event A ∈ F is given, that is, some collection of states is given, then the remaining states can also be identified and thus the complement Ac is also an event. The idea of a σ-field is to model a certain level of information. In particular, as the σ-field becomes larger, more and more events can be identified. We will later introduce a concept of an increasing flow of information, formally represented by an ordered (increasing) family of σ-fields. M. Rutkowski (USydney) Slides 5: Filtrations and Conditioning 6 / 37 Probability Measure Definition (Probability Measure) A map P : F → [0, 1] is called a probability measure if: 1 P (Ω) = 1, 2 for any sequence Ai, i ∈ N of pairwise disjoint events we have P Ai = P (Ai) . i∈N i∈N The triplet (Ω, F, P) is called a probability space. By convention, the probability of all possibilities is 1 (see 1). Probability should satisfy σ-additivity (see 2) Note that P (∅) = 0 and for an arbitrary event A ∈ F we have P (Ac) = 1 − P (A). M. Rutkowski (USydney) Slides 5: Filtrations and Conditioning 7 / 37 Example: σ-Fields Example (5.1) We take Ω= {ω1,ω2,ω3,ω4} and we define the σ-fields: F1 = {∅, Ω} F2 = ∅, Ω, {ω1,ω2} , {ω3} , {ω4} , {ω1,ω2,ω3} , {ω1,ω2,ω4} , {ω3,ω4} Ω F3 = 2 (the class of all subsets of Ω). Note that F1 ⊂F2 ⊂F3, that is, the information increases: F1: no information, except for the set Ω. F2: partial information, since we cannot distinguish between the occurrence of either ω1 or ω2. F3: full information, since {ω1}, {ω2}, {ω3} and {ω4} can be observed. M. Rutkowski (USydney) Slides 5: Filtrations and Conditioning 8 / 37 Example: Probability Measure Example (5.1 Continued) We define the probability measure P on the σ-field F2 2 1 1 P ({ω1,ω2})= , P ({ω3})= , P ({ω4})= . 3 6 6 The σ-additivity of P leads to P ({ω1,ω2}∪{ω3}∪{ω4})=1= P (Ω) . Ω Note that P is not yet defined on the σ-field F3 =2 and in fact the extension of P from F2 to F3 is not unique. For any α ∈ [0, 2/3] we may set 2 P ({ω1})= α = − P ({ω2}) . α 3 α M. Rutkowski (USydney) Slides 5: Filtrations and Conditioning 9 / 37 Partitions Definition Let I be some index set. Assume that we are given a collection (Bi)i∈I of subsets of Ω. Then the smallest σ-field containing this collection is denoted by σ ((Bi)i∈I ) and is called the σ-field generated by (Bi)i∈I . Definition (Partition) By a partition of Ω, we mean any collection P = (Ai)i∈I of non-empty subsets of Ω such that the sets Ai are pairwise disjoint, that is, Ai ∩ Aj = ∅ whenever i = j and i∈I Ai =Ω. Lemma A partition P = (Ai)i∈I generates a σ-field F if every set A ∈ F can be represented as follows: A = j∈J Aj for some subset J ⊂ I. M. Rutkowski (USydney) Slides 5: Filtrations and Conditioning 10 / 37 Partition Associated with a σ-Field Definition (Partition Associated with F) A partition of Ω associated with a σ-field F is a collection of non-empty sets Ai ∈ F for some i ∈ I such that 1 Ω= i∈I Ai. 2 The sets Ai are pairwise disjoint, i.e., Ai ∩ Aj = ∅ for i = j. 3 For each A ∈ F there exists J ⊆ I such that A = i∈J Ai. Lemma For any σ-field F of subsets of a finite state space Ω, a partition associated with this σ-field always exists and is unique. M. Rutkowski (USydney) Slides 5: Filtrations and Conditioning 11 / 37 Partition Associated with a σ-Field Further properties of partitions: If Ω is countable then for any σ-field F there exists a unique partition P of Ω associated with F. It is also clear that this partition generates F, so that F = σ(P). The sets Ai in a partition must be smallest, specifically, if F = σ(P) and A ∈ F is such that A ⊆ Ai then A = Ai. The probability of any A ∈ F equals the sum of probabilities of Ais in the partition generating F, specifically, A = Ai ⇒ P (A)= P (Ai) . i∈J i∈J M. Rutkowski (USydney) Slides 5: Filtrations and Conditioning 12 / 37 Example: Partition Associated with a σ-Field Example (5.2) Consider the σ-field F2 introduced in Example 5.1. The unique partition associated with F2 is given by P2 = {ω1,ω2} , {ω3} , {ω4} . Define the probabilities 2 1 1 P ({ω ,ω })= , P ({ω })= , P ({ω })= . 1 2 3 3 6 4 6 Then for each event A ∈ F2 the probability of A can be easily evaluated, for instance 5 P ({ω ,ω ,ω })= P ({ω ,ω })+ P ({ω })= . 1 2 4 1 2 4 6 M. Rutkowski (USydney) Slides 5: Filtrations and Conditioning 13 / 37 Random Variable Let F be an arbitrary σ-field of subsets of Ω. In the next definition, we do not assume that the sample space is discrete. Definition (F-Measurability) A map X :Ω → R is said to be F-measurable if for every closed interval [a, b] ⊂ R the preimage (i.e. the inverse image) under X belongs to F, that is, X−1([a, b]) := {ω ∈ Ω | X (ω) ∈ [a, b]} ∈ F. Equivalently, for any real number x X−1((−∞,x]) := {ω ∈ Ω | X (ω) ≤ x} ∈ F. If X is F-measurable then X is called a random variable on (Ω, F). M. Rutkowski (USydney) Slides 5: Filtrations and Conditioning 14 / 37 Partition and Random Variable Proposition (5.1) Let X :Ω → R be an arbitrary function. Let F be a σ-field and P = (Ai)i∈I be the unique partition associated with F. Then X is F-measurable if and only if for each Ai ∈ P there exists a constant ci such that X maps Ai to {ci}, that is, X (ω)= ci for all ω ∈ Fi. Proof of Proposition 5.1 (⇒). (⇒) Assume that (Ai)i∈I is a partition of the σ-field F and that X is F-measurable. Let j ∈ I be an index and let an element ω ∈ Aj be arbitrary. Define cj := X(ω). We wish to show that X(ω)= cj for all −1 ω ∈ Aj. Since X is F-measurable, X (cj) ∈ F. M. Rutkowski (USydney) Slides 5: Filtrations and Conditioning 15 / 37 Proof of Proposition 5.1 Proof of Proposition 5.1 (⇒). By properties 2. and 3. in the definition of a σ-field, we have −1 ∅ = X (cj) ∩ Aj ∈ F. It is obvious that the inclusion −1 X (cj) ∩ Aj ⊂ Aj holds. Therefore, from the aforementioned minimality property of the sets contained in the partition, we obtain −1 X (cj ) ∩ Aj = Aj. But this means that X(ω)= cj for all ω ∈ Aj and thus X is constant on Aj. By varying j, we obtain such cjs for all j ∈ I. M. Rutkowski (USydney) Slides 5: Filtrations and Conditioning 16 / 37 Proof of Proposition 5.1 Proof of Proposition 5.1 (⇐). (⇐) Assume now that X :Ω → R is a function, which is constant on all sets Aj belonging to the partition and that the cj are given as in the statement of the proposition. Let [a, b] be a closed interval in R. We define C := cj | j ∈ I and cj ∈ [a, b] . Since i∈I Ai =Ω no other elements then the cj occur as values of X. Therefore, −1 −1 −1 X ([a, b]) = X (C)= X (cj)= Aj ∈ F j |cj ∈C j |cj ∈C where the last equality holds by property 3.