# Toward categorical risk measure theory

## Takanori Adachi

We introduce a category that represents varying risk as well as ambiguity. We give a generalized conditional expectation as a presheaf for this category, which not only works as a traditional conditional expectation given a $\sigma$-field but also is compatible with change of measure. Then, we reformulate dynamic monetary value measures as a presheaf for the category. We show how some axioms of dynamic monetary value measures in the classical setting are deduced as theorems in the new formulation, which is evidence that the axioms are correct. Finally, we point out the possibility of giving a theoretical criteria with which we can pick up appropriate sets of axioms required for monetary value measures to be good, using a topology-as-axioms paradigm.

Keywords: conditional expectation, Radon-Nikodym derivative, monetary value measure, sheaf, Grothendieck topology

2010 MSC: Primary 91B30, 16B50; secondary 91B82, 18F10

Theory and Applications of Categories, Vol. 29, 2014, No. 14, pp 389-405.

Published 2014-08-05.

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