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Acta Applicandae Mathematicae

, Volume 100, Issue 1, pp 1–14 | Cite as

Optimisation of Measures on a Hyperfinite Adapted Probability Space

  • Sergio Albeverio
  • Frederik S. HerzbergEmail author
Article

Abstract

The minimisation problem for a functional \(P\mapsto u(\Gamma ,P,\tilde{g})\) is considered, where \(\tilde{g}\) is an ℝ n -valued stochastic process, defined on some filtered probability space \(\Gamma=(\Gamma,({\mathcal{G}}_{t})_{t\in[0,1]},\mathbb{P})\) , and P is an admissible probability measure in the sense that it obeys (1) some uniform equivalence condition with respect to the given measure ℙ on Γ, and (2) a finite number (possibly zero) of arbitrarily given other conditions that require the expectation (with respect to P) of some continuous bounded function φ of \((\tilde{g}_{t_{1}},\ldots,\tilde{g}_{t_{k}})\) , for t 1,…,t k ∈[0,1], to lie within some closed set. We assume that u can be formulated through finite compositions of conditional expectations and bounded continuous functions.

Under the assumption of |φ| being uniformly bounded from below and some condition on the dimension of \(\phi(\tilde{g}_{t_{1}},\ldots ,\tilde{g}_{t_{k}})\) , the existence of a solution on hyperfinite adapted probability spaces, as well as its minimality among admissible measures on any other adapted probability space, is proven. Also, a coarseness result for the Loeb operation is established.

The main result of this paper, however, is a “standard result”: It does not include any reference to nonstandard analysis and can be perfectly understood without any familiarity with nonstandard analysis.

Keywords

Optimisation in measure spaces Loeb measures Nonstandard analysis 

Mathematics Subject Classification (2000)

28E05 49J55 03H05 28A33 91B24 

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Copyright information

© Springer Science+Business Media B.V. 2007

Authors and Affiliations

  1. 1.Institut für Angewandte MathematikUniversität BonnBonnGermany
  2. 2.Institut für Mathematische WirtschaftsforschungUniversität BielefeldBielefeldGermany

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