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August 31 Math Colloquium

Renling Jin, CofC

An application of nonstandard analysis to the economic theory of insurance

An individual insurance agent can be viewed as a random variable on a probability space S. For a large number of insurance agents, the risk can be canceled out when these agents behave independently of each other. Suppose we want to establish a model of an insurance system with infinitely many agents. The indexes of these agents form another probability space T and the collection of these agents can be viewed as a random variable f(i,j) on a product of two probability spaces S times T. Naturally, this random variable should have (1) joint measurability on S times T and (2) for any two distinct j and j’ in T, the random variables f(*,j) and f(*,j’) are independent with respect to S. However, above two properties are inconsistent for any non-trivial cases. We will explain why they are inconsistent and introduce the work of Y. Sun which solves the triviality problem using nonstandard analysis. Nonstandard analysis will be introduced in detail. The audience are assumed to have learned calculus and have some knowledge of abstract probability theory.

 

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