5: FILTRATIONS and CONDITIONING [.03In]

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. of a σ-ﬁeld.

M. Rutkowski (USydney) Slides 5: Filtrations and Conditioning 17 / 37 PART 2

FILTRATIONS AND ADAPTED STOCHASTIC PROCESSES

M. Rutkowski (USydney) Slides 5: Filtrations and Conditioning 18 / 37 Information Flow: Filtration

In a typical application, the information about random events increases over time. To model the information flow, we use the concept of a filtration. The following definition covers the cases of the discrete and continuous time.

Deﬁnition (Filtration)

A family (Ft)0≤t≤T of σ-ﬁelds on Ω is called a ﬁltration if Fs ⊂ Ft whenever s ≤ t. For brevity, we denote F = (Ft)0≤t≤T .

We interpret the σ-ﬁeld Ft as the information available to an agent at time t. In particular, F0 represents the information available at time 0, that is, the initial information. We assume that the information accumulated over time can only grow, so that we never forget anything!

M. Rutkowski (USydney) Slides 5: Filtrations and Conditioning 19 / 37 Stochastic Process

Deﬁnition (Stochastic Process)

A family X = (Xt)0≤t≤T of random variables is called a stochastic process. A stochastic process X is said to be F-adapted if for every t = 0, 1,...,T the random variable Xt is Ft-measurable.

Example (5.3) Consider once again the elementary market model. The stochastic process of the stock price is S0 and S1 on Ω= {ω1,ω2} and the ﬁltration is F0 = {∅, Ω} and Ω F1 = ∅, Ω, {ω1}, {ω2} = 2 . Note that F0 is the initial information, which means that the investor knows only all possible states.

M. Rutkowski (USydney) Slides 5: Filtrations and Conditioning 20 / 37 Filtration Generated by a Stochastic Process

The initial information at time 0 is usually given by the trivial σ-ﬁeld F0 = {∅, Ω} since all prices are known at 0, so that there is no uncertainty.

Let X = (Xt)0≤t≤T be a stochastic process on the probability space (Ω, F, P). X For a fixed t, we define the σ-field Ft by setting

X −1 Ft = σ Xs ([a, b]) 0 ≤ s ≤ t, a ≤ b . X X Then the ﬁltration F = (Ft )0≤t≤T is called the ﬁltration generated by the process X.

M. Rutkowski (USydney) Slides 5: Filtrations and Conditioning 21 / 37 Example 5.4: Stock Price Model

S2 = 1.8 ω1 P(ω1) = 3/16 s9 sss sss sss S1 C = 1.3 ¦C KKK ¦¦ KKK ¦¦ KKK ¦¦ K% ¦ S2 = 1.2 ω2 P(ω2) = 5/16 ¦¦ ¦¦ ¦¦ ¦¦ S0 = 1 88 88 88 88 8 S2 = 1.1 ω3 P(ω3) = 4/16 88 9 8 sss 88 ss 8 sss 8 ss S1 = 0.7 KKK KKK KKK K% S2 = 0.5 ω4 P(ω4) = 4/16

M. Rutkowski (USydney) Slides 5: Filtrations and Conditioning 22 / 37 Example: Stock Price Model

Example (5.4 Continued) S F0 = {∅, Ω} since S0 is deterministic. At t = 1, we have

Ω if a ≤ 0.7 and b ≥ 1.3 −1  {ω1,ω2} if 0.7 < a ≤ 1.3 and b ≥ 1.3 S1 ([a, b]) =   {ω3,ω4} if a ≤ 0.7 and 0.7 ≤ b< 1.3 ∅ otherwise   and thus S F1 = ∅, Ω, {ω1,ω2} , {ω3,ω4} . S Ω F2 = 2 since the partition generating F2 is

P2 = {ω1}, {ω2}, {ω3}, {ω4} . M. Rutkowski (USydney) Slides 5: Filtrations and Conditioning 23 / 37 PART 3

CONDITIONAL EXPECTATION

M. Rutkowski (USydney) Slides 5: Filtrations and Conditioning 24 / 37 Conditional Expectation

Let (Ω, F, P) be a ﬁnite (or countable) probability space. Let X be an arbitrary F-measurable random variable. Assume that G is a σ-ﬁeld which is contained in F.

Let (Ai)i∈I be the unique partition associated with G.

Our next goal is to deﬁne the conditional expectation EP(X| G), that is, the conditional expectation of a random variable X with respect to a σ-ﬁeld G.

The expected value EP(X) will be obtained from EP(X| G) by setting G = F0, that is, EP(X)= EP(X| F0).

M. Rutkowski (USydney) Slides 5: Filtrations and Conditioning 25 / 37 Conditional Expectation

Deﬁnition (Conditional Expectation)

The conditional expectation EP(X| G) of X with respect to G is deﬁned as the random variable which satisﬁes, for every ω ∈ Ai, 1 EP(X| G)(ω)= X(ωl)P(ωl)= xk P(X = xk |Ai) P(Ai) ωl∈Ai xk where the summation is over all possible values of X and

−1 P(X = xk | Ai) = (P(Ai)) P({X = xk}∩ Ai)

is the conditional probability of the event {ω ∈ Ω | X(ω)= xk} given Ai. Hence 1 1 1 EP(X| G)= EP(X Ai ) Ai . P(Ai) i∈I

M. Rutkowski (USydney) Slides 5: Filtrations and Conditioning 26 / 37 Properties of Conditional Expectation

EP(X| G) is well deﬁned by equation and, by Proposition 5.1, the conditional expectation EP(X| G) is a G-measurable r.v.

EP(X| G) is the best estimate of X given the information represented by the σ-ﬁeld G. The following identity uniquely characterises the conditional expectation (in addition to G-measurability):

X(ω)P(ω)= EP(X| G)(ω)P(ω), ∀ G ∈ G. ω∈G ω∈G

One can represent this equality using (discrete) integrals: for every G ∈ G,

X dP = EP(X| G) dP, ∀ G ∈ G. G G

M. Rutkowski (USydney) Slides 5: Filtrations and Conditioning 27 / 37 Properties of Conditional Expectation

Proposition (5.2)

Let (Ω, F, P) be endowed with sub-σ-ﬁelds G and G1 ⊂ G2 of F. Then 1 Tower property: If X :Ω → R is an F-measurable r.v. then

EP(X| G1)= EP EP(X| G2) | G1 = EP EP(X| G1) | G2 . 2 Taking out what is known: If X :Ω → R is a G-measurable r.v. and Y :Ω → R is an F-measurable r.v. then

EP(XY | G)= X EP(Y | G).

3 Trivial conditioning: If X :Ω → R is an F-measurable r.v. independent of G then

EP(X| G)= EP(X).

M. Rutkowski (USydney) Slides 5: Filtrations and Conditioning 28 / 37 Example 5.5: Conditional Expectation

S: 2 = 9 ω1 P(ω1) = 6/25 3 u: 5 uu uu uu uu S1C = 8 C II 2 ¨ I 5 ¨ II ¨¨ II 2 ¨ II 5 ¨ $ ¨ S2 = 6 ω2 P(ω2) = 4/25 ¨¨ ¨¨ ¨¨ ¨¨ S0 =6 5 66 66 6 3 66 5 6 S: 2 = 6 ω3 P(ω3) = 6/25 6 2 u: 66 5 uu 6 uu 6 uu 6 uu S1 = 4 II 3 II 5 II II I$ S2 = 3 ω4 P(ω4) = 9/25

M. Rutkowski (USydney) Slides 5: Filtrations and Conditioning 29 / 37 Example: Conditional Expectation

Example (5.5)

The underlying probability space is given by Ω= {ω1,ω2,ω3,ω4}.

At time t = 0, the stock price is known and unique value is S0 = 5. S Hence the σ-ﬁeld F0 is the trivial σ-ﬁeld. At time t = 1, the stock can take two possible values and

Ω if a ≤ 4 and 8 ≤ b −1  {ω1,ω2} if 4 < a and 8 ≤ b S1 ([a, b]) =   {ω3,ω4} if a ≤ 4 and b< 8 ∅ if a< 4 and b< 4  S  so that F1 = {∅, {ω1,ω2}, {ω3,ω4}, Ω}. S Ω At time t = 2, we have F1 = 2 .

M. Rutkowski (USydney) Slides 5: Filtrations and Conditioning 30 / 37 Example: Conditional Expectation

Example (5.5 Continued)

6 P({ω ∈ A1} ∩ {S2(ω) = 9}) P(ω1) 25 3 P(S2 = 9 | A1)= = = = P(A ) P({ω ,ω }) 2 5 1 1 2 5 4 P({ω ∈ A1} ∩ {S2(ω) = 6}) P(ω2) 25 2 P(S2 = 6 | A1)= = = = P A P ω ,ω 2 ( 1) ({ 1 2}) 5 5 P({ω ∈ A1} ∩ {S2(ω) = 3}) P(∅) P(S2 = 3 | A1)= = = 0 P(A1) P({ω1,ω2}) P({ω ∈ A2} ∩ {S2(ω) = 9}) P(∅) P(S2 = 9 | A2)= = = 0 P(A2) P({ω3,ω4}) 6 P({ω ∈ A2} ∩ {S2(ω) = 6}) P(ω3) 25 2 P(S2 = 6 | A2)= = = = P A P ω ,ω 3 ( 2) ({ 3 4}) 5 5 9 P({ω ∈ A2} ∩ {S2(ω) = 3}) P(ω4) 25 3 P(S2 = 3 | A2)= = = = P A P ω ,ω 3 ( 2) ({ 3 4}) 5 5

M. Rutkowski (USydney) Slides 5: Filtrations and Conditioning 31 / 37 Example: Conditional Expectation

Example (5.5 Continued) We have

S 3 2 39 EP(S2 |F1 )(ω)=9 +6 +3 0= for ω ∈ A1 5 5 5 S 2 3 21 EP(S2 |F1 )(ω)=9 0+6 +3 = for ω ∈ A2 5 5 5 and thus S 39/5 if ω ∈{ω1,ω2} EP(S2|F1 )= 21/5 if ω ∈{ω3,ω4} Note that

S 39 2 21 3 141 EP EP(S2|F1 ) = + = 5 5 5 5 25 6 4 6 9 141 EP(S2)=9 +6 +6 +3 = 25 25 25 25 25

M. Rutkowski (USydney) Slides 5: Filtrations and Conditioning 32 / 37 PART 4

CHANGE OF A PROBABILITY MEASURE

M. Rutkowski (USydney) Slides 5: Filtrations and Conditioning 33 / 37 Change of a Probability Measure

Let P and Q be equivalent probability measures on (Ω, F). Let the Radon-Nikodym density of Q with respect to P be dQ (ω)= L(ω), P-a.s. dP meaning that L is F-measurable and, for every A ∈ F,

X dQ = XL dP. A A If Ω is ﬁnite then this equality becomes

X(ω) Q(ω)= X(ω)L(ω) P(ω). ω∈A ω∈A

The r.v. L is strictly positive P-a.s. and EP(L) = 1. Equality EQ(X)= EP(XL) holds for any Q-integrable random variable X (it suﬃces to take A =Ω).

M. Rutkowski (USydney) Slides 5: Filtrations and Conditioning 34 / 37 Abstract Bayes Formula

Lemma (5.1: Bayes Formula) Let G be a sub-σ-ﬁeld of F and let X be a Q-integrable random variable. Then the Bayes formula holds

EP(XL | G) EQ(X | G)= . EP(L | G)

Proof.

EP(L | G) is strictly positive so that the RHS is well deﬁned. By our assumption, the random variable XL is P-integrable. Therefore, it suﬃces to show that the equality

EP(XL | G)= EQ(X | G) EP(L | G)

holds for every random variable X.

M. Rutkowski (USydney) Slides 5: Filtrations and Conditioning 35 / 37 Proof of Lemma 5.1 for General σ-ﬁelds (MATH3975)

Proof. Since the RHS of the last formula deﬁnes a G-measurable random variable, it suﬃces to verify that for all sets G ∈ G, we have

XL dP = EQ(X | G)EP(L | G) dP. G G For every G ∈ G, we obtain

M. Rutkowski (USydney) Slides 5: Filtrations and Conditioning 36 / 37 Proof of Lemma 5.1 for Partitions (MATH3075)

Proof.

Now assume that G = σ(A1,...,Am) and Q(ωl)= L(ωl)P(ωl). Then on every event Ai 1 EQ(X| G)= X(ωl)Q(ωl) Q(Ai) ωl∈Ai 1 1 EP(LX| G)= L(ωl)X(ωl)P(ωl)= X(ωl)Q(ωl) P(Ai) P(Ai) ωl∈Ai ωl∈Ai

1 Q(Ai) EP(L| G)= L(ωl)P(ωl)= . P(Ai) P(Ai) ωl∈Ai

Hence on every event Ai from the partition P = {A1,...,Am}

EP(XL | G) EQ(X | G)= . EP(L | G)

This ends the proof of Lemma 5.1 when G = σ(A1,...,Am). M. Rutkowski (USydney) Slides 5: Filtrations and Conditioning 37 / 37