Toward categorical risk measure theory
type: Item
instance of: scholarly article
zbMATH Open document ID: 1304.91088
author: Takanori Adachi
publication date: 2014-08-25T00:00:00Z
full work available at URL: 29-14abs.html
Mathematics Subject Classification ID: 91B30
Mathematics Subject Classification ID: 18F10
Mathematics Subject Classification ID: 91G80
zbMATH DE Number: 6334032
zbMATH Keywords: axiomatic risk measure theory
zbMATH Keywords: dynamic monetary value measure
zbMATH Keywords: category theory
zbMATH Keywords: dynamic programming principle
zbMATH Keywords: time consistency
zbMATH Keywords: Grothendieck topology
MaRDI profile type: Publication
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title: Toward categorical risk measure theory
published in: Theory and Applications of Categories
review text: The author introduces a category that represents the time dependence of risk as well as the point-in-time coexistence of multiple subjective probability measures defining risk. The notion of dynamic monetary value measures is reformulated as a presheaf for the category, and it is shown that some axioms of dynamic monetary value measures in the classical setting can be deduced as theorems in the new formulation. The author argues that such results provide evidence that the axioms are appropriate, and points out the possibility of giving theoretical criteria with which one can isolate appropriate sets of axioms required for monetary value measures to be good, by finding an appropriate Grothendieck topology for which monetary value measures satisfying the given axioms become sheaves.NEWLINENEWLINELet \((\Omega,{\mathcal G},{\mathbb P})\) be a measure space. For a \(\sigma\)-field \({\mathcal F} \subseteq {\mathcal G}\), set \(L({\mathcal F}) = L^{\infty}(\Omega,{\mathcal F},{\mathbb P}|{\mathcal F})\). Let \(T > 0\) and fix a filtration \(\{{\mathcal G}_{t} : 0 \leq t \leq T \}\) on \((\Omega,{\mathcal G})\). A \textit{dynamic monetary value function} is a collection of functions \(\{\varphi_{t} : 0 \leq t \leq T\}\), \( \varphi_{t} : L({\mathcal G}_{T}) \rightarrow L({\mathcal G}_{t})\) \((0 \leq t \leq T)\), with the following properties: \begin{itemize} \item[-] Cash invariance: for every \(X \in L({\mathcal G}_{T})\) and \(Z \in L({\mathcal G}_{t})\), \(\varphi_{t}(X + Z) = \varphi_{t}(X) + Z\); \item[-] Monotonicity: for every \(X,Y \in L({\mathcal G}_{T})\), \(X \leq Y\) implies \(\varphi_{t}(X) \leq \varphi_{t}(Y)\); \item[-] Normalization: \(\varphi_{t}(0) = 0\). \end{itemize}NEWLINENEWLINEExamples of dynamic monetary value functions include the negative of Value-at-Risk on various horizons.NEWLINENEWLINEA dynamic monetary value function satisfies the \textit{dynamic programming principle} if for all \(0 \leq s \leq t \leq T\) and \(X \in L({\mathcal G}_{T})\),…
reviewed by: Tamás Mátrai
description: scientific article; zbMATH DE number 6334032