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slides_sde_2 - Class contents 1 Probability background 2...

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Class contents: 1. Probability background 2. Wiener process 3. Ito integral 12
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Probability Background A probability space is a triple (Ω , F , P ), where Ω is the set of outcomes, F is the set of events and P : F → [0 , 1] is a function that assigns probabilities to events satisfying cer- tain rules. 13
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Definition 1 (Measurable Space) If Ω is a given non empty set, then a σ -algebra F on Ω is a collection F of subsets of Ω that sat- isfy: (1) Ω ∈ F ; (2) F ∈ F ⇒ F c ∈ F , where F c = Ω F is the complement set of F in Ω; and (3) F 1 , F 2 , . . . ∈ F ⇒ + i =1 F i ∈ F . 14
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Definition 2 (Probability Measure) A prob- ability measure on , F ) is a set function P : F → [0 , 1] such that: (1) P ( ) = 0 , P (Ω) = 1; and (2) If A 1 , A 2 , . . . ∈ F are mutually disjoint sets then P + i =1 A i = + i =1 P ( A i ) . 15
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Question 1: Give an example of a probabil- ity space and distinguish clearly the events F ∈ F from the outcomes ω Ω. Question 2: Give an example of two dif- ferent σ -algebras, G ⊂ F for the same set of outcomes Ω. Can you give an intuitive interpretation of the relation G ⊂ F ? Question 3: Is the intersection of σ -algebras still a σ -algebra? Question 4: What about the union of σ - algebras? 16
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Definition 3 (generated σ -algebra) Given a family of sets, { A n } , there exists a unique σ -algebra, σ ( { A n } ) , s.t. 1. { A n } ⊂ σ ( { A n } ) , 2. if F is a σ algebra , { A n } ⊂ F ⇒ σ ( { A n } ) ⊂ F 17
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Definition 4 A random variable X , in the probability space , F , P ) , is a function X : Ω R d , such that the inverse image X 1 ( A ) ≡ { ω Ω : X ( ω ) A } ∈ F , for all open subsets A of R d . Equivalently, we may say that X is an F - measurable function and write X ∈ F . 18
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Example: Consider a finite family of disjoint sets, { A n } N n =1 and let Ω ≡ ∪ 1 n N A n , F ≡ σ ( { A n } ). What condition has to satisfy X : Ω R in order to be a random variable in (Ω , F )?
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