Lecture 4.pdf - Financial Engineering with Stochastic Calculus I Background in probability theory 40 62 Outline Construction of probability spaces

Lecture 4.pdf - Financial Engineering with Stochastic...

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Financial Engineering with Stochastic Calculus I 40 / 62 Background in probability theory September 10, 2019 Outline Construction of probability spaces Random variables and measurability
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Financial Engineering with Stochastic Calculus I 41 / 62 Background in probability theory Probability spaces Definition 3.1 A probability space consists of a set Ω , a family F of subsets of Ω ( σ -algebra ), and a function P ( probability measure ) which assigns to every A ∈ F a probability P [ A ] [0 , 1] . The σ -algebra F specifies those sets of outcomes for which we can assign a meaningful likelihood. Those sets are called events . In general, we cannot assign likelihood in a meaningful way to all such possible sets of outcomes, but in this course we will encounter only sets for which we are able to!
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Financial Engineering with Stochastic Calculus I 42 / 62 Background in probability theory For F we demand a) Ω ∈ F b) if A ∈ F , then the complement A c ∈ F c) if a sequence of events A 1 , A 2 , ... ∈ F , then j =1 A j ∈ F .
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