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Stochastic Analysis in Mathematical Finance MA5248 Lecture Notes 2 Conditional Expectations ¸1. Two Examples. Let (Ω, T, IP) be a probability space. We are going to introduce “conditional expectations”. Let us begin with some special cases and then proceed to two concrete examples (in items 3 and 6). 1. Let Λ ∈ T with IP(Λ) 0. Define IP Λ () on T as follows: IP Λ (E) = IP(Λ ∩ E) IP(Λ) . (1.1) It is trivial to check that IP Λ () thus defined in (1.
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  Stochastic Analysis in Mathematical FinanceMA5248Lecture Notes 2Conditional Expectations § 1. Two Examples. Let (Ω , F  , IP) be a probability space. We are going to introduce “ conditionalexpectations ”. Let us begin with some special cases and then proceed to twoconcrete examples (in items 3 and 6).1. Let Λ ∈ F  with IP(Λ) > 0. Define IP Λ ( · ) on F  as follows:IP Λ ( E  ) =IP(Λ ∩ E  )IP(Λ) . (1 . 1)It is trivial to check that IP Λ ( · ) thus defined in (1.1) is a probability measure (p.m.)on F  , which is known as the “conditional probability relative to Λ.” It is usuallywritten as IP( ·| Λ). Note also that (1.1) can also be written asIP(Λ ∩ E  ) = IP(Λ)IP Λ ( E  ) . The integral w.r.t. this p.m. is called the “conditional expectation relative to Λ”: EE  Λ [ X  ] =   Ω X  ( ω )IP Λ ( dω ) =1IP(Λ)   Λ X  ( ω )IP( dω ) = EE  [ X  ; Λ]IP(Λ) , (1 . 2)provided that EE  [ X  ] is defined, (i.e., EE  | X  | < ∞ ).(To establish (1.2), one only needs to make use of the “ standard machine ,” namely,to begin with X  as indicator r.v.’s; then simple r.v.’s; nonnegative r.v.’s; etc. Fordetails, see § 3.2 below.) Recall that EE  [ X  ; Λ] = EE  [ XI  Λ ] =   Λ X d IP . 2. However, if IP(Λ) = 0, we decree that IP Λ ( E  ) = 0 for every E  ∈ F  .3. We now give our first example. Example 1. Suppose there is a countable measurable partition { Λ n , n ≥ 1 } of Ω,namely:Ω = ∪ n Λ n , Λ n ∈ F  , Λ m ∩ Λ n = ∅ , if  m  = n. 1  Then we have, when EE  [ X  ] is defined, (i.e., EE  | X  | < ∞ ),IP( E  ) =  n IP(Λ n ∩ E  ) =  n IP(Λ n )IP Λ n ( E  ) EE  [ X  ] =  n   Λ n X  ( ω )IP( dω ) =  n IP(Λ n ) EE  Λ n [ X  ] , (1 . 3)Let G be the σ -algebra generated by this partition. Given an integrable r.v. X  ∈ F  ,we define the function  , denoted by EE  [ X  | G ] (or EE  G [ X  ]), on Ω by: for ω ∈ Ω, EE  [ X  | G ]( ω ) =  n EE  Λ n [ X  ] I  Λ n ( ω ) . Observe that EE  [ X  | G ] is a discrete r.v. that assumes the value EE  Λ n [ X  ] = EE  [ X  ; Λ n ]IP(Λ n )on the event Λ n for each n . More precisely, the above function EE  [ X  |G ] can beexpressed as EE  [ X  | G ]( · ) =  n EE  Λ n [ X  ] I  Λ n ( · ) =  n EE  [ X  ; Λ n ]IP(Λ n ) I  Λ n ( · ) . (1 . 4)Obviously, this function EE  [ X  | G ] is measurable with respect to the sub- σ -algebra G , i.e., EE  [ X  | G ] ∈ G .Now (1.3) can be re-written as EE  [ X  ] =   Ω X d IP =  n   Λ n EE  [ X  |G ] d IP =   Ω EE  [ X  |G ] d IP . (1 . 5)Furthermore, if Λ ∈ G , then Λ is a union of a sub-collection of the Λ n ’s, and notethat the following also holds: ∀ Λ ∈ G : EE  [ X  ; Λ] = EE  [ XI  Λ ] =   Λ X d IP =   Λ EE  [ X  | G ] d IP . (1 . 6)(Clearly, (1.5) is a special case of (1.6) with Λ being replaced by Ω.)4. Remarks: (a) Look at (1.6) and especially note that the last two terms are integrals over thesame event Λ ∈ G ; but the integrand X  of the 3rd term is measurable w.r.t. F  ,2  while the integrand EE  [ X  |G ] on the far right is measurable w.r.t. the sub- σ -algebra G .(b) If Λ in (1.6) is replaced by another event in F \G , then (1.6) may not hold anymore. In other words, (1.6) holds in general for Λ ∈ G only .(c) We now claim that for any integrable X  , EE  [ X  | G ] is unique a.s. That is, if thereis another r.v. W  ∈ G such that ∀ Λ ∈ G :   Λ X d IP =   Λ W d IP , (1 . 7)then W  = EE  [ X  |G ] a.s. Proof of the Claim. PutΛ = { ω : EE  [ X  | G ]( ω ) > W  ( ω ) } . Obviously, Λ ∈ G , and Λ = ∪ n Λ n , whereΛ n = { ω : EE  [ X  | G ]( ω ) > W  ( ω ) + n − 1 } . If IP(Λ) > 0, then there exists an n ≥ 1 such that IP { Λ n } > 0. On this Λ n ,   Λ n X d IP =   Λ n W d IP <   Λ n EE  [ X  | G ] d IP =   Λ n X d IP , a contradiction. So, IP(Λ) = 0 and EE  [ X  | G ] ≤ W  a.s. In a similar fashion, byinterchanging EE  [ X  | G ] and W  above we arrive at W  ≤ EE  [ X  | G ] a.s., and henceconclude that W  = EE  [ X  |G ] a.s., which is the claim.5. We have therefore proved that the EE  [ X  | G ] in (1.6) is unique up to an equivalence.From now on we will also use EE  [ X  | G ] to denote the corresponding equivalenceclass, and call any particular member of the class a “ version ” of the conditionalexpectation relative to the σ -algebra G . Before proceeding, let us summarize what we have obtained. For a countable mea-surable partition { Λ n n ≥ 1 } , let G = σ { Λ n n ≥ 1 } ⊂ F  . For X  ∈ F  , EE  [ X  | G ] ∈ G and it is unique a.s.; and if  X  has finite mean, then EE  [ X  | G ] has also finite mean(by(1.5)).6. Now the second example. Example 2. Let Z  be a discrete r.v. taking distinct values a 1 , a 2 , a 3 , ... . PutΛ n = { Z  = a n } . Then we have a countable measurable partition { Λ n , n ≥ 1 } of 3  Ω. It is obvious to see that σ ( Z  ) = σ { Λ n , n ≥ 1 } , denoted by G . By what hasbeen mentioned above in the first example, EE  [ X  |G ] is defined for any X  ∈ F  with EE  | X  | < ∞ , and then we can define the conditional expectation relative to ther.v. Z  to be EE  [ X  | Z  ] def. = EE  [ X  | G ] = EE  [ X  | σ ( Z  )] . Obviously, the expectation of  X  may change given that the event Λ n has occurred.Indeed, if Q(Λ) def. = IP { Λ | Λ n } , it makes more sense to calculate EE  Q[ X  ] than it does to calculate EE  IP [ X  ]. ( EE  IP {·} denotes expectation with respect to the probability measure IP.)An important observation that we would like to mention is that: EE  [ X  | Z  ] is nowa function of  Z  , i.e., there is a Borel measurable function ϕ such that EE  [ X  | Z  ] = ϕ ( Z  ) a.s.More precisely, define ϕ ( n ) = EE  [ X  ; Λ n )IP(Λ n )= EE  [ X  ; Z  = n ]IP( Z  = n ) . (1 . 8)For ω ∈ Λ n = { Z  = n } , by referring to (1.4), EE  [ X  | Z  ]( ω ) = EE  [ X  ; Λ n ]IP(Λ n )= EE  [ X  ; Z  = n ]IP( Z  = n )= ϕ ( n ) = ϕ ( Z  ( ω )) . (1 . 9)Equivalently, EE  [ X  | Z  ]( ω ) = ϕ ( Z  )( ω ) =  n EE  [ X  ; Z  = n )IP( Z  = n ) I  { n } ( Z  ( ω )) . (1 . 10)This fact will be mentioned again in § 2.4 below.7. Remark. Observe that EE  [ X  | Z  ] ∈ σ ( Z  ) , i.e., EE  [ X  | Z  ] is a measurable w.r.t. σ ( Z  ). By Doob-Dynkin Lemma (in § 2.5 of Notes 1), we know that EE  [ X  | Z  ] is now a function of  Z  , i.e., there is a Borelmeasurable function ϕ such that EE  [ X  | Z  ] = ϕ ( Z  ) a.s.4
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