Random optimization on random sets

Lépinette, Emmanuel

doi:10.1007/s00186-019-00686-6

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2020

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“…The notion of a random set gives meaning to random objects X whose realizations X(ω), ω ∈ Ω, take values as subsets of some space X ; see [2]. These objects have an important role in mathematical finance and stochastic optimization; see e.g., [6][7][8][9][10]. By considering the Boolean valued model associated with the underlying probability space, we show that a random set corresponds to a Borel set in this model.…”

Section: Introductionmentioning

confidence: 99%

Boolean Valued Representation of Random Sets and Markov Kernels with Application to Large Deviations

López

García

2020

Mathematics

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We establish a connection between random set theory and Boolean valued analysis by showing that random Borel sets, random Borel functions, and Markov kernels are respectively represented by Borel sets, Borel functions, and Borel probability measures in a Boolean valued model. This enables a Boolean valued transfer principle to obtain random set analogues of available theorems. As an application, we establish a Boolean valued transfer principle for large deviations theory, which allows for the systematic interpretation of results in large deviations theory as versions for Markov kernels. By means of this method, we prove versions of Varadhan and Bryc theorems, and a conditional version of Cramér theorem.

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