Continuous Time - Probability Theory

Continuous Time - Probability Theory - 1 Temple University...

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Unformatted text preview: 1 Temple University Continuous Time Finance Probability Theory 2 Continuous Time Finance Probability Theory 3 Probability Theory Discrete probability theory deals with events that occur in countable sample spaces. a probability distribution is called discrete if it is characterized by a probability mass function . Continuous probability theory deals with events that occur in a continuous sample space the term "continuous probability distribution" is reserved for distributions that have probability density functions . 4 Probability Theory Measure-theoretic probability theory : unifies the discrete and the continuous, and makes the difference a question of which measure is used Allows distributions that are neither discrete nor continuous (nor mixtures of the two) Measure generalizes notions such as "length", "area", and "volume" (but not all of its applications have to do with physical sizes). Informally, given some base set, a "measure" is any consistent assignment of "sizes" to (some of) the subsets of the base set. 5 Probability and Measure Foundations of the Theory of Probability " by Andrey Nikolaevich Kolmogorov is the foundation of modern probability theory. Andrey Kolmogorov 1903-1987 = = i i E P E E P P E P ) ( ) ( E disjoint Countable 1 ) ( ) ( 2 1 i ( ( 6 Probability Theory 1.1 Probability Spaces 7 Sample Space Sample Space of an experiment or random trial is the set of all possible outcomes. Any (measurable) subset of the sample space is usually called an event. Sample space = {2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12} = = HTHHTH e.g., outcome possible 8 sigma-algebra -algebra (or sigma-algebra) over a set is i. a nonempty collection F of subsets of that is ii. closed under complementation and iii. (closed under ) countable unions of its members. -algebra (or sigma-algebra) over a set provided that i. The empty set belongs to F ii. Whenever A belongs to F, its complement A c also belongs iii. Whenever sequence A n belongs to F, so does their union: . = 1 n n A If we have a -algebra, all the operations we might want to do to the sets will give us other sets in the -algebra 9 Set Theory De Morgans Laws : not (A and B) = (not A) or (not B) not (A or B) = (not A) and (not B) Augustus De Morgan 1806-1871 He formulated De Morgan's laws and introduced the term mathematical induction c C c c C c B A B A B A B A = = ) ( ) ( = = = 1 1 n n c n c n A A 10 Probability Space Probability Space : formalizes three interrelated ideas : First, a random sample point (outcome of experiment, state of nature, possibility etc.) F : Second, an event, --- something that will occur or not, depending on the revealed sample point....
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Continuous Time - Probability Theory - 1 Temple University...

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