Toward categorical risk measure theory (Q2877681)

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 \(X \in L({\mathcal G}_{T})\), \(\varphi_{s}(X) = \varphi_{s}(\varphi_{t}(X))\) holds; and it satisfies \textit{time consistency} if for all \(0 \leq s \leq t \leq T\) and \(X,Y \in L({\mathcal G}_{T})\), \(\varphi_{t}(X) \leq \varphi_{t}(Y)\) implies \(\varphi_{s}(X) \leq \varphi_{s}(Y)\). These properties are usually introduced as axioms for dynamic monetary value functions; see, e.g. [\textit{K. Detlefsen} and \textit{G. Scandolo}, Finance Stoch. 9, No. 4, 539--561 (2005; Zbl 1092.91017)].NEWLINENEWLINEThe main result of the paper is that in a categorical setting, the counterparts of the dynamic programming principle and time consistency properties can be obtained as corollaries. Let \(\chi = \chi(\Omega,{\mathcal G})\) be the set of all pairs \(({\mathcal F},{\mathbb Q})\), where \({\mathcal F} \subseteq {\mathcal G}\) is a \(\sigma\)-field and \({\mathbb Q}\) is a probability measure on \((\Omega,{\mathcal G})\). For \({\mathcal U} = ({\mathcal F},{\mathbb Q}) \in \chi\), we introduce \({\mathcal F}_{\mathcal U} = {\mathcal F}\) and \({\mathbb Q}_{\mathcal U} = {\mathbb Q}\). Then \(\chi\) becomes a category with exactly one arrow defined by \(\star_{\mathcal U}^{\mathcal V}: {\mathcal V} \rightarrow {\mathcal U}\) being in \(\chi\) if and only if \({\mathcal F}_{\mathcal V} \subseteq {\mathcal F}_{\mathcal U}\) and \({\mathbb Q}_{\mathcal U} \ll {\mathbb Q}_{\mathcal V}\), where \(\ll\) stands for ``being absolute continuous to''.NEWLINENEWLINESet \(L_{\mathcal U} = L^{\infty}(\Omega,{\mathcal F}_{\mathcal U},{\mathbb Q}_{\mathcal U}|_{{\mathcal F}_{\mathcal U}})\) and \([X]_{\mathcal U} = \{Y \in L^{\infty}(\Omega,{\mathcal F}_{\mathcal U}) : Y \sim_{{\mathbb Q}_{\mathcal U}} X\}\) \((X \in L^{\infty}(\Omega,{\mathcal F}_{\mathcal U}))\), and consider the functor \(L : \chi \rightarrow \text{Set}\), defined by NEWLINE\[CARRIAGE_RETURNNEWLINE\begin{tikzcd} \mathcal V \ar[r] \ar[d] & L_{\mathcal V} \ar[d, "L_{\mathcal U}^{\mathcal V}"] & \\ {\mathcal U}\ar[r] & L_{\mathcal U} \rlap{\,,}\end{tikzcd}CARRIAGE_RETURNNEWLINE\]NEWLINE where \(L_{\mathcal U}^{\mathcal V} : [X]_{\mathcal V}\mapsto [X]_{\mathcal U}\). With this notation, a \textit{monetary value measure} is a contravariant functor \(\varphi : \chi^{\text{op}} \rightarrow \text{Set}\) satisfying the following two conditions:NEWLINE\begin{itemize}NEWLINE\item[(1)] for \({\mathcal U} \in \chi\), \(\varphi({\mathcal U}) = L_{\mathcal U}\);NEWLINE\item[(2)] for \({\mathcal V} \rightarrow {\mathcal U}\) in \(\chi\), the map \(\varphi_{\mathcal U}^{\mathcal V} = \varphi({\mathcal V} \rightarrow {\mathcal U})\), a function from \(L_{\mathcal U}\) to \(L_{\mathcal V}\), satisfies: NEWLINE\begin{itemize} \item [-] Cash invariance: for every \(X \in L_{\mathcal U}\) and \(Z \in L_{\mathcal V}\), \(\varphi_{\mathcal U}^{\mathcal V}(X + L_{\mathcal U}^{\mathcal V}(Z)) = \varphi_{\mathcal U}^{\mathcal V}(X) + Z\) \({\mathbb Q}_{\mathcal V}\)-a.e.; \item [-] Monotonicity: for every \(X,Y \in L_{\mathcal U}\), \(X \leq Y\) implies \(\varphi_{\mathcal U}^{\mathcal V}(X) \leq \varphi_{\mathcal U}^{\mathcal V}(Y)\) \({\mathbb Q}_{\mathcal V}\)-a.e.; \item [-] Normalization: if \({\mathbb Q}_{\mathcal V} = {\mathbb Q}_{\mathcal U}\) then \(\varphi_{\mathcal U}^{\mathcal V}(0_{L_{\mathcal U}}) = 0_{L_{\mathcal V}}\) \({\mathbb Q}_{\mathcal V}\)-a.e. \end{itemize} NEWLINE\end{itemize}NEWLINENEWLINEThe main result of the paper is that a monetary value measure \(\varphi : \chi^{\text{op}} \rightarrow \text{Set}\) also has the following properties: if \({\mathcal W} \rightarrow {\mathcal V} \rightarrow {\mathcal U}\) are arrows in \(\chi\) then \({\mathbb Q}_{\mathcal V} = {\mathbb Q}_{\mathcal U}\) implies \(\varphi^{\mathcal W}_{\mathcal U} = \varphi^{\mathcal W}_{\mathcal U} \circ L_{\mathcal U}^{\mathcal V} \circ \varphi_{\mathcal U}^{\mathcal V}\) (dynamic programming principle); for every \(X,Y \in L_{\mathcal U}\), \(\varphi_{\mathcal U}^{\mathcal V}(X) \leq \varphi_{\mathcal U}^{\mathcal V}(Y)\) implies \(\varphi_{\mathcal U}^{\mathcal W}(X) \leq \varphi_{\mathcal U}^{\mathcal W}(Y)\) (time consistency). In this sense, these properties are natural axioms in the classical setting. The author also proposes a program to identify another set of axioms the appropriateness of which can be established this way.