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Search results with tag "Measurable ran dom variables"
Stochastic Calculus: An Introduction with Applications
www.math.uchicago.edu;F;P) is a probability space and Yis an integrable random variable. Suppose Gis a sub ˙-algebra of F. Then E[Y jG] is de ned to be the unique (up to an event of measure zero) G-measurable random variable such that if A2G, E[Y1 A] = E[E[Y jG]1 A]: Uniqueness follows from the fact that if Z 1;Z 2 are G-measurable ran-dom variables with E[Z 1 1 A ...
Stochastic Calculus: An Introduction with Applications
www.math.uchicago.edu2 are G-measurable ran-dom variables with E[Z 1 1 A] = E[Z 2 1 A] for all A2G, then PfZ 1 6= Z 2g= 0. Existence of the conditional expectation can be deduced from the Radon-Nikodym theorem. Let (A) = E[Y1 A]. Then is a (signed) measure on (;G;P) with ˝P, and hence there exists an L1 random variable Zwith (A) = E[Z1 A]