数理解析研究所講究録
第 1818 巻 2012 年 163-180

163
A Note on Risk Measure Theory
from a Category-Theoretic Point of View
Takanori Adachi
Graduate School of International Corporate Strategy
Hitotsubashi University
Abstract

We reformulate dynamic monetary value measures with the language of category theory. We show some axioms in the old setting are deduced as theorems in the new formulation, which may be one of the evidences that the axioms are natural. We also demonstrate a
topology-as-axioms paradigm in order to give a theoretical criteria with which we can pickup appropriate sets of axioms required for monetary value measures to be good.

1

Introduction

The risk measure theory we are formulating is a theory of dynamic (multi-period) monetaryrisk measures. Since the axiomatization of monetary risk measures was initiated by [ADEH99],
many axioms such as law invariance have been presented ([KusOl], [FSII]). Especially after
introducing multi-period (or dynamic) versions of monetary risk measures, a lot of investigations have been made so far $[ADE^{+}07]$ . Those investigations are valuable in both theoreticaland practical senses. However, it may be expected to have some theoretical criteria of picking
appropriate sets of axioms out of them. Thinking about the recent events such as the CDS hedging failure at Morgan Chase, the importance of selecting appropriate axioms of monetaryrisk measures becomes even bigger than before. In this note, we formalize dynamic monetaryrisk measures in the language of category theory in order to add a new view point to the risk
measure theory.
Category theory is an area of study in mathematics that examines in an abstract way theproperties of maps (called morphisms or arrows) satisfying some basic conditions. It has been
applied in many fields including geometry, logic, computer science and string theory. Even formeasure theory, there are some attempts to apply category theory such as [Jac06] or [Bre77].
However, in finance theory, as far as we know, there has been nothing. We will use it forforrnulating dynamic rnonetary risk measures.
In this note, we will stress two points. One is how we can formulate some concepts ofdynamic risk measure theory in the language of category theory and show some axioms in theold setting become theorems in our setting. Another point is to present a criteria of selecting setsof axioms required for monetary value measure theory in a sheaf-theoretic point of view.
The remainder of this paper consists of four sections.
$JP$2000 subject classifcations: Primary $91B30,16B50$; secondary $91B82,18F10$
subject classifications: C02, D46
Keywords and phrases: monetary risk measure, monetary value measure, category, sheaf, Grothendieck topology

$AMS$
$JEL$

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In Section 2, we provide brief reviews about dynamic risk measure theory and category

theory.
In Section 3, we give a definition of monetary value measures as contravariant functors
from a set of -fields as a poset. Then, we will see the resulting monetary value measuressatisfy time consistency condition and dynamic programming principle that were introduced
as axioms in the old version of dynamic risk measure theory.
In Section 4, we $wm$ investigate a possibility of finding an appropriate Grothendieck topology for which monetary value measures satisfying given axioms become sheaves. We also
introduce the notion of complete set of axioms with which we give a method to constmct amonetary value measure satisfying the axiom from any given monetary value measure.
In Section 5, we investigate the situation of monetary value measures in a quite simple casethat accepts concave monetary value$\Omega=\{1,2,3\}$ , and show that any set of axioms overmeasures is not complete.
$\sigma$$\Omega$

2 Review of Dynamic Risk Measures and Category Theories
In this section we give a very brief review of dynamic risk measure theory and category theory.

Throughout this note, all discussions are under the probability space$(\Omega,\mathcal{F},\mathbb{P})$.

2.1 Dynamic Risk Measure Theory
First, we review the case of one period monetary risk measures.

Definition 2.1. one period monetary risk measure is a function
the following axioms
$A$

.

$0$

.

$\rho$:$L^{p}(\Omega,\mathcal{F},\mathbb{P})arrow \mathbb{R}$satisfying

Cash invariance: $(\forall X)(\forall a\in \mathbb{R})\rho(X+a)=\rho(X)-a,$Monotonicity: $(\forall X)(\forall Y)X\leq Y\Rightarrow\rho(X)\geq\rho(Y)$ ,Normalization: $\rho(0)=0,$

where
bounded by the$L^{p}(\Omega,\mathcal{F},\mathbb{P})$is the space of equivalence classes of$\Vert\cdot\Vert_{p}$$\mathbb{R}$-valued random variables which are

norm.

Here are examples of one period risk measures.

Example 2.2. [One Period Monetary Risk Measures]
1. Value at Risk
$VaR_{\alpha}(X):=\inf\{m\in \mathbb{R}|\mathbb{P}(X+m<0)\leq\alpha\}$2. Expected shortfall$ES_{\alpha}(X) :=\frac{1}{1-\alpha}\int_{\alpha}^{1}VaR_{u}(X)du$165
Now, we will define the notion of dynamic monetary risk measures. However, we actuallyadopt the way of using a monetary value measure instead of using a monetary risk measure
below by conforming the manner in recent literature such as $[ADE^{+}07]$ and [KM07], where wehave a relation $\varphi(X)=-\rho(X)$ for any possible scenario (i.e. a random variable)
From now on, we think a monetary value measure instead of a monetary risk measuredefined by$:=-\rho(X)$ .
$\varphi$$\rho$$X.$$\varphi$$\rho$$\varphi(X)$Definition 2.3. For a -field, is the space of all equivalenceclasses of bounded -valued random variables, equipped with the usual $\sup$ norm.
$\mathcal{U}\subset \mathcal{F},$$\sigma$$L(\mathcal{U})$$:=L^{\infty}(\Omega,\mathcal{U},\mathbb{P}|\mathcal{U})$$\mathbb{R}$Definition 2.4. Let
collection of functions$\mathbb{F}=\{\mathcal{F}_{t}\}_{t\in[0,T]}$.

.
.

Cash invariance:

Monotonicity:
Normalization:$\varphi=\{\varphi_{t} :be a filtration.

$A$

L(\mathcal{F}_{T})arrow L(\mathcal{F}_{t})\}_{t\in[0,T]}$dynamic monetary value measure is asatisfying$(\forall X\in L(\mathcal{F}_{T}))(\forall Z\in L(\mathcal{F}_{t}))\varphi_{t}(X+Z)=\varphi_{t}(X)+Z,$,$(\forall X\in L(\mathcal{F}_{T}))(\forall X\in L(\mathcal{F}_{T}))X\leq Y\Rightarrow\varphi_{t}(X)\leq\varphi_{t}(Y)$$\varphi_{t}(0)=0.$

Note that the directions of some inequalities in Definition 2.1 are different from those of
Definition 2.4 because we now monetary value measures instead of monetary risk measures.
Since dynamic monetary value measures treat multi-period situations, we may require
some extra axioms to regulate them toward the time dimension. Here are two possible suchaxioms.

Axiom 2.5. [Dynamic programming principle] For $0\leq s\leq t\leq T,$$\varphi_{s}(\varphi_{t}(X))$.

Axiom 2.6. [Time consistency] For $0\leq s\leq t\leq T,$$\varphi_{s}(X)\leq\varphi_{s}(Y)$2.2$(\forall X\in L(\mathcal{F}_{T}))\varphi_{s}(X)=$$(\forall X,\forall Y\in L(\mathcal{F}_{T}))\varphi_{t}(X)\leq\varphi_{t}(Y)$$\Rightarrow$.

Category Theory

The description about category theory presented in this subsection is very limited. For those
who are interested in more detail about category theory, please consult [Mac97].

Definition 2.7. [Categories] categoryof arrows or morphisms such that
$A$

$C$

consists of a collection$\mathcal{O}_{C}$of objects and a collection$\mathcal{M}_{C}$1. there are tw functions$0$$\mathcal{M}_{C\vec{\frac{dom}{cod}}}\mathcal{O}_{C}.$

When $dom(f)=A$ and cod $(f)=B$ , we write $f$ : $Aarrow B.$
We define a so-called $hom$ -set of given objects and by $Hom_{C}(A, B)$ $:=\{f\in \mathcal{M}c|f$ :$Aarrow B\}$ . We sometimes write $C(A,B)$ for $Hom_{C}(A,B)$ .
$A$

2. for : $Aarrow B$ andand $f.$$f$$g$: $Barrow C$ , there is an arrow$B$$g\circ f$: $Aarrow C$ , called the composition of

$g$

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3. every object$A$

is associated with an identity arrow $1_{A}:Aarrow A$ satisfying $f\circ 1_{A}=f$ and$1_{A}\circ g=g$where $dom(f)=A$ and cod $(g)=A.$

Example 2.8. [Examples of Categories].

1. Set: the category of small sets$\mathcal{O}_{Set}$$:=$

collection of all small sets,collection of all functions between small sets.

$:=$$o\mathcal{M}_{Set}$

2. Top : the category of topological spaces..

$\mathcal{O}_{Top}$$:=$$\mathcal{M}_{Top}$

collection of all topological spaces,$:=$collection of all continuous functions between topological spaces.

3. Opposite category

.

Let

$C$

$0$

$C^{op}$be a given category. Then we define its opposite category$C^{op}$by the following way.

$\mathcal{O}_{C^{op}}:=\mathcal{O}_{C\prime}$

for $A,$$B\in \mathcal{O}_{C},$$Hom_{C^{op}}(A, B)$$:=Hom_{C}(B, A)$.

Example 2.9. [Partial Ordered Sets as Categories]
A partial ordered set (sometimes we call it poset) $(S, \leq)$ can be considered as a category
defined in the following way.

.

.

$\mathcal{O}_{S}:=S,$

for $a,b\in S,$ $Hom_{S}(a,b)$$:=\{\begin{array}{ll}\{i_{b}^{a}\} if a\leq b,\emptyset otherwise.\end{array}$

We see the correspondence between definitions of posets and categories below.

1. Reflexivity vs. identity arrows: $a\leq a$$aarrow^{1_{a}=i_{l}^{a}}$$a$2. Transitivity vs. composition arrows: $a\leq b$ and $b\leq c\Rightarrow a\leq c$$ab\underline{i_{b}^{a}}$$P_{c}=i_{c}^{b}oi_{b}^{l} A\mathcal{C}$Definition 2.10. [Functors] Let

$C$

and

$\mathcal{D}$be two categories. Afunctor

functions,$F_{\mathcal{O}}$:$\mathcal{O}_{C}arrow \mathcal{O}_{\mathcal{D}}$and$F_{\mathcal{M}}$:$\mathcal{M}_{C}arrow \mathcal{M}_{\mathcal{D}}$satisfying$F:Carrow \mathcal{D}$consists of two

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1. $f:Aarrow B\Rightarrow F(f):F(A)arrow F(B)$,

2. $F(g\circ f)=F(g)\circ F(f)$ ,3. $F(1_{A})=1_{F(A)}.$
Definition 2.11. [Contravariant functors] functor :if two conditions 1 and 2 in Definition 2.10 are replaced by
$A$

$F$

$C^{op}arrow \mathcal{D}$is called a contravariantfunctor.

1. $f:Aarrow B\Rightarrow F(f):F(B)arrow F(A)$ ,2. $F(g\circ f)=F(f)\circ F(g)$ .

Example 2.12. [Contravariant Functor]$C^{op}\underline{Hom_{C}(-\prime C)}$Set

$f\downarrow AB Hom_{C}(A,C),\ni\uparrow Hom_{C}(f^{C)} g_{g}^{\circ}fI$$Hom_{C}(B,C)\ni$

Definition 2.13. [Natural Transformations] Let$C\vec{\frac{F}{G}}\mathcal{D}$mation : $Farrow G$ consists of a family of arrows
commute:$\alpha$be two functors.

$\langle\alpha_{C}|C\in \mathcal{O}_{C}\rangle$$A$

natural transfor-making the following diagram$C_{1} F(C_{1})arrow^{\alpha_{C_{1}}}G(C_{1})$$f\downarrow F(f)| \downarrow G(f)$$C_{2} F(C_{2})\overline{\alpha_{C_{2}}}G(C_{2})$Definition 2.14. [Functor Categories] Let
category such that.
.

$\mathcal{O}_{D^{C}}$

$:=$

$C$

collection of all functors from

$Hom_{\mathcal{D}^{C}}(F, G)$

$:=$

and

$C$

to

$\mathcal{D}$be categories. Afunctor category$\mathcal{D}^{C}$

is the$\mathcal{D},$collection of all natural transformations from$F$

to $G.$

3 Monetary Value Measures
Now we start defining monetary value measures with the language of category theory. First,
we introduce a simple category that is actually a partially ordered set derived by the -field$\sigma$

Definition 3.1. [Category ]$\chi$$\mathcal{F}.$168-fields of . Then, it becomes a poset with the setbe the set of all
1. Letinclusion relation . Moreover, as shown in Example 2.9, becomes a category whose
is defined by$hom$ setfor$\chi$$sub-\sigma$$:=\chi(\mathcal{F})$$\mathcal{F}$$\chi$$\subset$$\mathcal{U},\mathcal{V}\in\chi$$Hom_{\chi}(\mathcal{V},\mathcal{U})$(3.1)$Hom_{\chi}(\mathcal{V},\mathcal{U}):=\{\begin{array}{ll}\{i_{u}^{\mathcal{V}}\} if \mathcal{V}\subset \mathcal{U},\emptyset otherwise.\end{array}$

The arrow$2.$

is called an inclusion map.

$i_{\mathcal{U}}^{\mathcal{V}}$$\perp;=\{\Omega,\emptyset\}$, which is the least element of$\chi.$All the discussions below depend on this particular category . But, you may notice inmany cases that we can replace with more restricted collections of -fields such as the fullof whose objects are -complete, or a totally ordered subset of which is
subcategory
considered as (strictly increasing) filtration.
, which is actually
We restrict the space of random variables bounded in the norm
necessary when showing the local property in Proposition 3.5. But, in many places, you canwith an arbitrary $p(p\geq 1)$ instead ofrelax it to$\chi$$\sigma$$\chi$$\mathbb{P}$$\chi$$\chi$$\chi_{c}$$a$$\Vert\cdot\Vert_{\infty}$$\infty.$$\Vert\cdot\Vert_{p}$Definition 3.2. [Monetary Value Measures]$A$monetary value measure is a contravariant functor$\varphi:\chi^{op}arrow$Set

satisfying the following two conditions:
1. for
2. for$\mathcal{U}\in\chi,$..
.

$\varphi(\mathcal{U})$$\mathcal{U},\mathcal{V}\in\chi$$:=L(\mathcal{U})$such that

Cash invariance:
Monotonicity:
Normalization:,$\mathcal{V}\subset \mathcal{U}$, the map$\varphi_{u}^{\mathcal{V}}$$:=\varphi(i_{u}^{\mathcal{V}})$: $L(u)arrow L(\mathcal{V})$ satisfies$(\forall X\in L(\mathcal{U}))(\forall Z\in L(\mathcal{V}))\varphi_{\mathcal{U}}^{\mathcal{V}}(X+Z)=\varphi_{u}^{\mathcal{V}}(X)+Z,$$(\forall X\in L(\mathcal{U}))(\forall Y\in L(\mathcal{U}))X\leq Y\Rightarrow\varphi_{u}^{\mathcal{V}}(X)\leq\varphi_{u}^{\mathcal{V}}(Y)$,$\varphi_{\mathcal{U}}^{\mathcal{V}}(0)=0.$

At this point, we do not require the monetary value measures to satisfy some familiar conditions such as concavity or law invariance. Instead of doing so, we want to see what kind of
properties are deduced from this minimal setting.
One of the key points of Definition 3.2 is that is a contravariant functor. So, for any triple
in , we have, as seeing in Diagram 3.1,
of -fields$\varphi$$\mathcal{W}\subset \mathcal{V}\subset \mathcal{U}$$\sigma$$\chi$$\varphi_{u}^{u}=1_{L(U)}$andExample 3.3. [Entropic Value Measure] Let
Set defined by$\lambda$.

$\varphi_{\mathcal{V}}^{\mathcal{W}}\circ\varphi_{u}^{\mathcal{V}}=\varphi_{u}^{\mathcal{W}}$(3.2)

be a non-zero real number. Then the functor$\varphi:\chi^{op}arrow$$\varphi_{\mathcal{U}}^{\mathcal{V}}(X) :=\lambda^{-1}\log \mathbb{E}^{\mathbb{P}}[e^{\lambda X}|\mathcal{V}]$where$\mathcal{V}\subset \mathcal{U}$in$\chi$

and$X\in L(\mathcal{U})$is a monetary value measure.

(3.3)

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$\chi^{op}arrow^{\varphi}$Set$\mathcal{W}\mapsto\varphi(\mathcal{W}) = L(\mathcal{W})$$i_{\mathcal{U}}^{\mathcal{W}}(_{\downarrow i_{\mathcal{U}}^{\mathcal{V}}1\varphi(i_{\mathcal{U}}^{\mathcal{V}})=}^{11\varphi(i_{\mathcal{V}}^{\mathcal{W}})=}\mathcal{V}\mapsto\varphi(\mathcal{V})=i_{\mathcal{V}}^{\mathcal{W}}.\cdot\cdot\varphi_{\mathcal{V}}^{\mathcal{W}}|\varphi_{\mathcal{U}}^{\mathcal{V}}|L(\mathcal{V}))\varphi_{\mathcal{U}}^{\mathcal{W}}$$\mathcal{U}\mapsto\varphi(\mathcal{U}) = L(\mathcal{U})$Diagram 3.1
Definition 3.4. [Concave Monetary Value Measure]concave if for any
inand$\mathcal{V}\subset \mathcal{U}$$\chi,$$X,Y\in L(\mathcal{U})$$A$

monetary value measure$\varphi$

is said to be$\lambda\in[0,1],$

.

(3.4)

$\varphi_{u}^{\mathcal{V}}(\lambda X+(1-\lambda)Y)\geq\lambda\varphi_{\mathcal{U}}^{\mathcal{V}}(X)+(1-\lambda)\varphi_{u}^{\mathcal{V}}(Y)$An entropic value measure is concave.

Here are some properties of monetary value measures.

Proposition 3.5. Let
1.

$\varphi:\chi^{op}arrow$Set be a monetary value measure, andbe$\sigma$-fields in$\chi.$$(\forall X\in L(\mathcal{V}))\varphi_{\mathcal{U}}^{\mathcal{V}}(X)=X,$

2. Idempotentness:
3. Local property:$1_{A^{c}}\varphi_{\mathcal{U}}^{\mathcal{V}}(Y)$$(\forall X\in L(\mathcal{U}))\varphi_{\mathcal{U}}^{\mathcal{V}}(\varphi_{\mathcal{U}}^{\mathcal{V}}(X))=\varphi_{\mathcal{U}}^{\mathcal{V}}(X)$,

5. Time consistency:$(\forall X\in L(\mathcal{U}))\varphi_{u}^{\mathcal{W}}(X)=\varphi_{u}^{\mathcal{W}}(\varphi_{\mathcal{U}}^{\mathcal{V}}(X))$,$(\forall X\in L(\mathcal{U}))(\forall Y\in L(\mathcal{U}))\varphi_{u}^{\mathcal{V}}(X)\leq\varphi_{u}^{\mathcal{V}}(Y)\Rightarrow\varphi_{u}^{\mathcal{W}}(X)\leq\varphi_{\mathcal{U}}^{\mathcal{W}}(Y)$

1. By cash invariance and normalization,

2. Since,$(\forall X\in L(\mathcal{U}))(\forall Y\in L(\mathcal{U}))(\forall A\in \mathcal{V})\varphi_{\mathcal{U}}^{\mathcal{V}}(1_{A}X+I_{A^{c}}Y)=1_{A}\varphi_{\mathcal{U}}^{\mathcal{V}}(X)+$

4. Dynamic programming principle:

Proof.

$\mathcal{W}\subset \mathcal{V}\subset \mathcal{U}$$\varphi_{\mathcal{U}}^{\mathcal{V}}(X)\in L(\mathcal{V})$$\varphi_{\mathcal{U}}^{\mathcal{V}}(X)=\varphi_{\mathcal{U}}^{\mathcal{V}}(0+X)=\varphi_{u}^{\mathcal{V}}(0)+X=X.$, it is obvious by 1.

3. First, we show that for any$A\in \mathcal{V},$.

$I_{A}\varphi_{\mathcal{U}}^{\mathcal{V}}(X)=I_{A}\varphi_{\mathcal{U}}^{\mathcal{V}}(]1_{A}X)$

Since

.

$X\in L^{\infty}(\Omega,\mathcal{U},\mathbb{P})$, we have$|X|\leq\Vert X\Vert_{\infty}$. Therefore,$1_{A}X-I_{A^{c}}\Vert X\Vert_{\infty}\leq 1_{A}X+]1_{A^{c}}X\leq I_{A}X+1_{A^{c}}\Vert X\Vert_{\infty}.$

(3.5)

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Then, by cash invariance and monotonicity,$\varphi_{u}^{\mathcal{V}}(]1_{A}X)-I_{A^{c}}\Vert X\Vert_{\infty}=\varphi_{\mathcal{U}}^{\mathcal{V}}(I_{A}X-1_{A^{c}}\Vert X\Vert_{\infty})$$\leq\varphi_{u(X)}^{\mathcal{V}}$$\leq\varphi_{u}^{\mathcal{V}}(1_{A}X+I_{A^{c}}\Vert X\Vert_{\infty})=\varphi_{u}^{\mathcal{V}}(I_{A}X)+I_{A^{c}}\Vert X\Vert_{\infty}.$

Then,$I_{A\varphi_{\mathcal{U}}^{\mathcal{V}}(I_{A}X)=1_{A}(\varphi_{u(1_{A}X)-I_{A^{c}}\Vert X\Vert_{\infty})}^{\mathcal{V}}}$$\leq 1_{A}\varphi_{\mathcal{U}}^{\mathcal{V}}(X)$$\leq 1_{A}(\varphi_{u}^{\mathcal{V}}(1_{A}X)+]1_{A^{c}}\Vert X\Vert_{\infty})=I_{A}\varphi_{u}^{\mathcal{V}}(1_{A}X)$.

Therefore, we get (3.5).

Next by using (3.5) twice, we have$\varphi_{u(1_{A}X}^{\mathcal{V}}+I_{A^{c}}Y)=I_{A\varphi_{u}^{\mathcal{V}}}(1_{A}X+I_{A^{c}}Y)+n_{A^{c}\varphi_{u}^{\mathcal{V}}}(n_{A}x+1_{A^{c}}Y)$$=]1_{A}\varphi_{u}^{\mathcal{V}}(1_{A}(1_{A}X+I_{A^{c}}Y))+I_{A^{c}}\varphi_{u}^{\mathcal{V}}(I_{A^{C}}(1_{A}X+1_{A^{c}}Y))$$=I_{A\varphi_{u}^{\mathcal{V}}}(1_{A}X)+1_{A^{c}}\varphi_{u}^{\mathcal{V}}(I_{A^{c}}Y)$$=u_{A\varphi_{uu}}^{\nu_{(X)+]1_{A^{c}}\varphi}\nu_{(Y)}}.$

4. By 2 and (3.2), we have.

$\varphi_{u}^{\mathcal{W}}(X)=\varphi_{\mathcal{V}}^{\mathcal{W}}(\varphi_{u}^{\mathcal{V}}(X))=\varphi_{\mathcal{V}}^{\mathcal{W}}(\varphi_{u}^{\mathcal{V}}(\varphi_{u}^{\mathcal{V}}(X)))=(\varphi_{\mathcal{V}}^{\mathcal{W}}\circ\varphi_{\mathcal{U}}^{\mathcal{V}})(\varphi_{\mathcal{U}}^{\mathcal{V}}(X))=\varphi_{u}^{\mathcal{W}}(\varphi_{u}^{\mathcal{V}}(X))$

5. Assume$\varphi_{\mathcal{U}}^{\mathcal{V}}(X)\leq\varphi_{u}^{\mathcal{V}}(Y)$. Then, by monotonicity and (3.2),$\varphi_{u(X)=\varphi_{\mathcal{V}}^{\mathcal{W}}(\varphi_{u(X))}^{\mathcal{V}}}^{\mathcal{W}}\leq\varphi_{\mathcal{V}}^{\mathcal{W}}(\varphi_{\mathcal{U}}^{\mathcal{V}}(Y))=\varphi_{u(Y)}^{\mathcal{W}}.$$\square$

In Proposition 3.5, two properties, dynamic programming principle and time consistency
are usually introduced as axioms ([DS06]). But, we derive them naturally here from the fact thatthe monetary value measure is a contravariant functor. This may be seen as another evidence
that the two axioms are quite natural.

4 Monetary Value Measures as Sheaves
In general, a contravariant functor : $C^{op}arrow$ Set is called a presheaf for a category . By
definition, a monetary value measure is a presheaf. The name presheaf suggests that it is related
to another concept sheaves, which is a quite important concept in some classical branches inmathematics such as algebraic topology. [MM92]. So, what makes a presheaf be a sheaf?
For a given set, a topology defined on it provides a criteria to distinguish good ( continuous) functions from given functions on the set. In a similar way, there is a concept called$C$$\rho$$=$

171

Figure 4.1: topology-as-axioms paradigm

a Grothendieck topology defined on a given category that gives a criteria to distinguish goodpresheaves ( sheaves) from given presheaves on the category. In both cases, (Grothendieck)topology can be seen as a vehicle to identify good functions (presheaves) among general functions (presheaves).
On the other hand, if we have a set of functions that we want to make good ( continuous),
we can find the weakest topology that makes the functions continuous. In a similar way, if wehave a set of presheaves that we want to make good, it is known that we can pick a Grothendiecktopology with which the presheaves become sheaves. See Figure 4.1 for the analogy.
Since a monetary value measure is a presheaf, if we have a set of good monetary value
measures ( the monetary value measures that satisfy a given set of axioms), we may find a
Grothendieck topology with which the monetary value measures become sheaves. We will see
a concrete shape of the Grothendieck topology in Section 4.1.
Now suppose we have a weak topology that makes given functions continuous. This, however, does not imply the fact that any continuous function w.r. . the topology is contained in
the originally given functions. Similarly, Suppose that we have a Grothendieck topology thatmakes all monetary value measures satisfying a given set of axioms sheaves. It, however, does
not mean that any sheaf w.r. . the Grothendieck topology satisfies the given set of axioms. We
will investigate this situation in Section 4.2.
$=$

$a$

$=$

$=$

$t$

$t$

4.1

A Grothendieck Topology as Axioms

In this subsection, we see a concrete shape of the Grothendieck topology with which all mone-tary value measures satisfying a given set of axioms become sheaves.
First, we review two concepts of Grothendieck typologies and sheaves.
Definition 4.1. Let
$1.$

2.

$\mathcal{U}\in\chi.$$\downarrow \mathcal{U}:=\{\mathcal{V}\in\chi|\mathcal{V}\subset \mathcal{U}\}.$$A$

sieve on$\mathcal{U}$is a set

3. For a sieve on$I$$\mathcal{U}$$I\subset\downarrow \mathcal{U}$and$\mathcal{V}\subset \mathcal{U}$4. Afamily of is an element$I$such thatin$\chi,$$(\forall \mathcal{V}\in\downarrow \mathcal{U})(\forall \mathcal{W}\in\downarrow\mathcal{U})[\mathcal{W}\subset \mathcal{V}\in I\Rightarrow \mathcal{W}\in I].$

$I\downarrow \mathcal{V}$$:=I\cap\downarrow \mathcal{V}.$$X\in\prod_{\mathcal{V}\in I}L(\mathcal{V})$. We write$X=(X_{\mathcal{V}})_{\mathcal{V}\in I}.$

172
5.

$A$

family$X=(X_{\mathcal{V}})_{\mathcal{V}\in\iota}$

is called a -martingale if$\mathbb{P}$$(\forall \mathcal{V}\in I)(\forall \mathcal{W}\in J)[\mathcal{W}\subset \mathcal{V}\Rightarrow \mathbb{E}^{\mathbb{P}}[X_{\mathcal{V}}|$$\mathcal{W}]=X_{\mathcal{W}}].$

A sieve on$\mathcal{U}$is considered as a kind of a time domain. We sometimes call a family a subpro-

cess.
Note thatis a sieve on$I\downarrow \mathcal{V}$$\mathcal{V}.$Set is a contravariant functor such that for1. :Definition4.2.
foris the set of all sieves on , and that$E$

2. A Grothendieck topology on:$\mathcal{V}arrow u$

in$\chi,$.

is a subfunctor $\int>E$ satisfying the following conditions:$\chi$,$( \forall \mathcal{U}\in\chi)\downarrow \mathcal{U}\in\int(\mathcal{U})$(b)$I\in E(\mathcal{U})$$E(i_{\mathcal{U}}^{\mathcal{V}})(I)=I\downarrow \mathcal{V}$$\mathcal{U}$$E(\mathcal{U})$(a)$i_{\mathcal{U}}^{\mathcal{V}}$$\chi^{op}arrow$$(\forall \mathcal{U}\in\chi)(\forall I\in 1(\mathcal{U}))(\forall K\in E(\mathcal{U}))[(\forall \mathcal{V}\in I)K\downarrow \mathcal{V}\in 1(\mathcal{V})\Rightarrow K\in J(\mathcal{U})].$

We say a sieve I 1-covers$\mathcal{U}$if$I\in 1(\mathcal{U})$.

is considered as a time horizon of a time domain if it is covered byHere is a well-known property of Grothendieck topologies.
$I$$\mathcal{U}$$I.$

Theorem 4.3. Let $\{\int_{a}|a\in A\}$ be a collection of Grothendieck topologies on . Then the subfunctor
is a Grothendieck topology. We write this 1 by
$J>E$ defined by$\chi$$f(\mathcal{U})$$\bigcap_{a\in A}J_{a}.$$:= \bigcap_{a\in A}1_{a}(\mathcal{U})$

Next we introduce concepts of families depending on a monetary value measure.

Definition 4.4. Let$\varphi\in Set^{\chi^{op}}$

1.

$A$

family2.

$A$random variable$X=(X_{\mathcal{V}})_{\mathcal{V}\in\int}$

be a monetary value measure, and be a sieve on$I$

is called -matching if$\mathcal{U}\in\chi.$$(\forall \mathcal{V}\in I)(\forall \mathcal{W}\in I)\varphi_{\mathcal{V}}^{\mathcal{V}\wedge \mathcal{W}}(X_{\mathcal{V}})=\varphi_{\mathcal{W}}^{\mathcal{V}\wedge \mathcal{W}}(X_{\mathcal{W}})$$\varphi$$\overline{X}\in L(U)$is called a -amalgamation for a family$X$$\varphi$$=(X_{\mathcal{V}})_{\mathcal{V}\in 1}$

if$(\forall \mathcal{V}\in I)\varphi_{u}^{\mathcal{V}}(X^{-})=X_{\mathcal{V}}.$The next two propositions give us some intuition about the relation between two concepts
just introduced, -matching and -amalgamation.
$\varphi$$\varphi$

be a monetary value measure, be a sieve on
Proposition 4.5. Let
be a family that has a -amalgamation. Then, is -matching.
$I$$\varphi\in Set^{\chi^{op}}$$X$$\varphi$

Proof. Let
any$X\in L(\mathcal{U})$$\nu,\mathcal{W}\in I,$$\mathcal{U}\in\chi$and$X=(X_{\mathcal{V}})_{\mathcal{V}\in I}$$\varphi$

be a -amalgamation. Then, for any$\mathcal{V}\in I,$$X_{\mathcal{V}}=\varphi_{u}^{\mathcal{V}}(X^{-})$$\varphi$. Therefore, for.

$\varphi_{\mathcal{V}}^{\mathcal{V}\wedge \mathcal{W}}(X_{\mathcal{V}})=X_{\mathcal{V}\wedge \mathcal{W}}=\varphi_{\mathcal{W}}^{\mathcal{V}\wedge \mathcal{W}}(X_{\mathcal{W}})$$\square$Proposition 4.6. Let
be a -matching family.

$\varphi\in Set^{l^{p}}$be a monetary value measure, be a sieve on$I$$\varphi$

1. For
2. If

Proof.

$\mathcal{W},\mathcal{V}\in I$$\mathcal{U}\in I,$

1.

$Xu$

,

if

$\mathcal{W}\subset \mathcal{V}$, we have$\varphi_{\mathcal{V}}^{\mathcal{W}}(X_{\mathcal{V}})=X_{\mathcal{W}}.$is the unique -amalgamation for X.
$\varphi$$\varphi_{\mathcal{V}}^{\mathcal{W}}(X_{\mathcal{V}})=\varphi_{\mathcal{V}}^{\mathcal{V}\wedge \mathcal{W}}(X_{\mathcal{V}})=\varphi_{\mathcal{W}}^{\mathcal{V}\wedge \mathcal{W}}(X_{\mathcal{W}})=\varphi_{\mathcal{W}}^{\mathcal{W}}(X_{\mathcal{W}})=X_{\mathcal{W}}.$$\mathcal{U}\in\chi$and$X=(X_{\mathcal{V}})_{\mathcal{V}\in\iota}$.

173
2. By 1,$x_{u}$

is a -amalgamation for X.
$\varphi$

Now let $X\in L(U)$ be another -amalgamation for X. Then for every
Put. Then, we have$\varphi$$\mathcal{V}$$:=\mathcal{U}$$\mathcal{V}\in I,$$X_{\mathcal{V}}=\varphi_{\mathcal{U}}^{\mathcal{V}}(X^{-})$.

$X_{\mathcal{U}}=\varphi_{\mathcal{U}}^{\mathcal{U}}(X^{-})=1u(X^{-})=\overline{X}.$$\square$Now we are at the position where we can introduce the concept of sheaves.

Definition 4.7. Let 1 be a Grothendieck topology onmonetary value measure
is called a sheaf (for ) if for any, any -covering sieve
and any -matchingfamilyhas a unique -amalgamation.
$\chi.$$J$$X=(X_{\mathcal{V}})_{\mathcal{V}\in I},$$\mathcal{U}\in\chi$

$X$

$A$

$\varphi\in Set^{\chi^{op}}$$I\in 1(\mathcal{U})$$J$$\varphi$$\varphi$

In the rest of this subsection, we will try to find a Grothendieck topology $for$ which a givenclass of monetary value measures specified by a given set of (extra) axioms are sheaves.
Let us consider a sieve on
as a subfunctor $I>Hom_{\chi}(-,\mathcal{U})$ , that is, a contravariant functor :Set defined by
$I$

$I$

$\mathcal{U}\in\chi$$\chi^{op}arrow$(4.1)$I(\mathcal{V}):=\{\begin{array}{ll}\{i_{\mathcal{U}}^{\mathcal{V}}\} if \mathcal{V}\in I,\emptyset if \mathcal{V}\not\in I.\end{array}$for$\mathcal{V}\in\chi.$Actually, by this convention, we can identify a -matching subprocess X on a sieve witha natural transformation X :$I$$\varphi$$Iarrow\varphi.$

The following theorem assures the existence of a Grothendieck topology making a given
monetary value measure a sheaf.
Proposition 4.8. Let$\varphi\in Set^{\chi^{op}}$be a monetary value measure, and define a subfunctorby$J_{\varphi}>\Rightarrow\Xi$$I\downarrow v-\downarrow \mathcal{V}$(4.2)$J_{\varphi}(\mathcal{U}):=\{I\in E(\mathcal{U})|(\forall \mathcal{V}\in\downarrow \mathcal{U})\forall_{x\downarrow_{A}} \exists!_{Y} \}$$\varphi$for$\mathcal{U}\in\chi$. Then, the subfunctor$J_{\varphi}$is the largest Grothendieck topology for which$\varphi$

is a sheaf.

Proof. Refer Example 3. $2.14c$ in [Bor94].

$\square$

By combining Proposition 4.8 and Theorem 4.3, we have the following corollary
Corollary 4.9. Let
be the collection of all monetary value measures satisfying a given setof axioms. Then, there exists a Grothendieck topology for which all monetary value measures in aresheaves, where the topology is largest among topologies rep resenting the axioms. We write the topology
by$\mathcal{M}\subset Set^{\chi^{op}}$$\mathcal{M}$$J_{\mathcal{M}}.$

Proof. Let$1_{\mathcal{M}}$$:= \bigcap_{\varphi\in \mathcal{M}}J_{\varphi}$etary value measure in$\mathcal{M}$. Then, it is the largest Grothendieck topology for which every monis a sheaf.
$\square$174
4.2 Complete sets of Axioms
Let be a fixed set of axioms. Then, for a given arbitrary monetary value measure , can we
make a good altemative for it? In other words, can we fin a monetary value measure thatsatisfies and is the best approximation of the original ? This is the theme of this subsection.
$\varphi$$\mathcal{A}$$d$$\mathcal{A}$$\varphi$

to be a full subcategory whose
For a Grothendieck topology 1 on , definein theobjects are all sheaves for J. Then, it is well-known that there exists a left adjoint$Sh(\chi,f)\subset Set^{\chi^{0\rho}}$$\chi$$\pi_{1}$following diagram.
$Sh(x,1)Set^{l^{p}}\overline{\overline{\pi_{1}}}$(4.3)$(\rfloor) (\rfloor)$$\pi_{1}(\varphi)-\varphi$

is well-known with the name sheafification functor, which comes with the folThe functor
lowing limit cone:$\pi_{1}$$\cdots-Nat(I, \varphi)Nat(K, \varphi)\underline{Nat(i_{l’}^{K}\varphi)}-\ldots$(4.4)

for sieves $I,K$ on
Theorem 4.10.

$\mathcal{U}$. It also satisfies the following theorem.

1.

$\pi_{1}(\varphi)$

is a sheaffor 1.

2. If is a sheaffor 1, then for any$\mathcal{U}\in\chi,$$\pi_{1}(\varphi)(\mathcal{U})\simeq L(\mathcal{U})$$\varphi$.

Theorem 4.10 suggests that for an arbitrary monetary value measure, the sheafification
functor provides one of its closest monetary value measures that may satisfy the given set of
axioms. To make this certain, we need a following definition.
Definition 4.11. Let

1.

$\mathcal{M}(\mathcal{A})$2.

$\mathcal{M}_{0}$3.

$\mathcal{A}$

$:=$

$:=$

$\mathcal{A}$be a set of axioms for monetary value measures.

the collection of all monetary value measures satisfying$\mathcal{A}.$the collection of all monetary value measures.

is called complete if$\pi_{J_{\mathcal{M}(A)}}(\mathcal{M}_{0})\subset \mathcal{M}(\mathcal{A})$

.

(4.5)

By Theorem 4.10, we have the following main result.
be a complete set of axioms. Then, for a monetary value measure
Theorem 4.12. Let
is the monetary value measure that is the best approximation satisfying axioms$\mathcal{A}$$\varphi\in \mathcal{M}0,$$\mathcal{A}.$$\pi_{J_{\mathcal{M}(A)}(\varphi)}$Now, we want to expect that some of the well-known sets of axioms such as those forconcave monetary value measures are complete. However, we will see a counterexample in

Section 5 in a quite simple case.

175
5 Completeness Condition on $\Omega=\{1,2,3\}$In this section, we investigate if the set of axioms of concave monetary value measures is com-plete in the case $\Omega=\{1,2,3\}$ with a -field$\sigma$

5.1$\mathcal{F}:=2^{\Omega}.$

Some Consideration on the Shape of Monetary Value Measures onFirst, we enumerate all possible sub$-\sigma$-fields of$\Omega$$\Omega$, that is, the shape of the category $\chi=\chi(\Omega)$which is like following:$\mathcal{U}_{1}\nearrow_{\mathcal{U}_{2}}^{\mathcal{U}_{\infty}}|\backslash _{\mathcal{U}_{3}}$(5.1)$\backslash |\mathcal{U}_{0}\nearrow$where$\mathcal{U}_{\infty}:=\mathcal{F}:=2^{\Omega},$$\mathcal{U}_{1}:=\{\emptyset, \{1\}, \{2,3\},\Omega\},$$\mathcal{U}_{2}:=\{\emptyset, \{2\}, \{1,3\},\Omega\},$$\mathcal{U}_{3}:=\{\emptyset, \{3\}, \{1,2\},\Omega\},$$u_{0}:=\{\emptyset,\Omega\}$.

The Banach spaces derived by the elements of(5.2)$\chi$

are:$L_{\infty}:=L:=L(\mathcal{U}_{\infty})=\{(a,b,c)|a,b,c\in \mathbb{R}\},$$L_{1}:=L(\mathcal{U}_{1})=\{(a,b,b)|a,b\in \mathbb{R}\},$$L_{2}:=L(\mathcal{U}_{2})=\{(a,b,a)|a,b\in \mathbb{R}\},$$L_{3}:=L(\mathcal{U}_{3})=\{(a,a,c)|a,c\in \mathbb{R}\},$$L_{0}:=L(\mathcal{U}_{0})=\{(a,a,a)|a\in \mathbb{R}\}$

Then, a monetary value measure$\varphi$:$\chi^{op}arrow$Set on$\chi$.

(5.3)

is determined by the following six

functions:$L_{\infty}$$\nearrow^{\varphi_{\infty}^{1}}|\varphi_{\infty}^{2}\backslash ^{\varphi_{\infty}^{3}}$$L_{1} L_{2} L_{3}$$\backslash _{\varphi_{1}^{0}}|\varphi\nearrow^{20\varphi_{3}^{0}}$$L_{0}$

We will investigate its concrete shape one by one by considering axioms it satisfies.

(5.4)

176
For

$\varphi_{\infty}^{1}$:$L_{\infty}arrow L_{1}$, we have by the cash invariance axiom,$\varphi_{\infty}^{1}(a,b,c)=\varphi_{\infty}^{1}((0,b-c,0)+(a,c,c))$$=\varphi_{\infty}^{1}((0,b-c,0))+(a,c,c)$$=(f_{12}(b-c),f_{11}(b-c),f_{11}(b-c))+(a,c,c)$$=(f_{12}(b-c)+a,f_{11}(b-c)+c,f_{11}(b-c)+c)$:
define mne functions

where$f_{11},f_{12}$$\mathbb{R}arrow \mathbb{R}$are defined by $(f_{12}(x),f_{11}(x),f_{11}(x))=\varphi_{\infty}^{1}(0,x,O)$ . Similarly, if we:by$f_{11},f_{12},f_{21},f_{22},f_{31},f_{32},g_{1},g_{2},g_{3}$$\mathbb{R}arrow \mathbb{R}$$(f_{12}(x),f_{11}(x),f_{11}(x))=\varphi_{\infty}^{1}(0,x,0)$,$(f_{21}(x),f_{22}(x),f_{21}(x))=\varphi_{\infty}^{2}(0,0,x)$,$(f_{31}(x),f_{31}(x),f_{32}(x))=\varphi_{\infty}^{3}(x,0,0)$,$(g_{1}(x),g_{1}(x),g_{1}(x))=\varphi_{1}^{0}(x,0,0)$,$(g_{2}(x),g_{2}(x),g_{2}(x))=\varphi_{2}^{0}(0,x,0)$,$(g_{3}(x),g_{3}(x),g_{3}(x))=\varphi_{3}^{0}(0,0,x)$

.

(5.5)

We can represent the original six functions in (5.4) by the mine functions defined in (5.5).
$\varphi_{\infty}^{1}(a,b,c)=(f_{12}(b-c)+a,f_{11}(b-c)+c,f_{11}(b-c)+c)$,$\varphi_{\infty}^{2}(a,b,c)=(f_{21}(c-a)+a,f_{22}(c-a)+b,f_{21}(c-a)+a)$ ,$\varphi_{\infty}^{3}(a,b,c)=(f_{31}(a-b)+b,f_{31}(a-b)+b,f_{32}(a-b)+c)$ ,$\varphi_{1}^{0}(a,b,b)=(g_{1}(a-b)+b,g_{1}(a-b)+b,g_{1}(a-b)+b)$ ,$\varphi_{2}^{0}(a,b,a)=(g_{2}(b-a)+a,g_{2}(b-a)+a,g_{2}(b-a)+a)$ ,$\varphi_{3}^{0}(a,a,c)=(g_{3}(c-a)+a,g_{3}(c-a)+a,g_{3}(c-a)+a)$

.

(5.6)

Next by the normahzation axiom, we have.

$f_{11}(0)=f_{12}(0)=f_{21}(0)=f_{22}(0)=f_{31}(0)=f_{32}(0)=g_{1}(0)=g_{2}(0)=g_{3}(0)=0$

Now suppose that we can partially differentiate the function
ments. Then, we have$\frac{\partial}{\partial a}\varphi_{\infty}^{1}(a,b,c)=(1,0,0)$(5.7)

in all three argu-$\varphi_{\infty}^{1}(a, b,c)$,$\frac{\partial}{\partial b}\varphi_{\infty}^{1}(a,b,c)=(f_{12}’(b-c),f_{11}’(b-c),f_{11}’(b-c))$,$\frac{\partial}{\partial c}\varphi_{\infty}^{1}(a,b,c)=(-f_{12}’(b-c),1-f_{11}’(b-c),1-f_{11}’(b-c))$.

Therefore, by the monotonicity, we have $f_{12}’(x)=0$ and $0\leq f_{11}’(x)\leq 1$ . Then by (5.7), we have$f_{12}(x)=0$ . Hence, for all
for all$x\in \mathbb{R},$$x\in \mathbb{R},$$f_{12}(x)=f_{22}(x)=f_{32}(x)=0$

.

(5.8)

177
With this knowledge, let us redefine the three functions$f_{1},f_{2},f_{3}$$(0,f_{1}(x),f_{1}(x))=\varphi_{\infty}^{1}(0,x,0)$,$(f_{2}(x),0,f_{2}(x))=\varphi_{\infty}^{2}(0,0,x)$,$(f_{3}(x),f_{3}(x),0)=\varphi_{\infty}^{3}(x,0,0)$.

,:$\mathbb{R}arrow \mathbb{R}$

by

(5.9)

Then, we have a new representation of the original six functions in (5.4):$\varphi_{\infty}^{1}(a,b,c)=(a,f_{1}(b-c)+c,f_{1}(b-c)+c)$ ,$\varphi_{\infty}^{2}(a,b,c)=(f_{2}(c-a)+a,b,f_{2}(c-a)+a)$,,$\varphi_{\infty}^{3}(a,b,c)=(f_{3}(a-b)+b,f_{3}(a-b)+b,c)$$\varphi_{1}^{0}(a,b,b)=(g_{1}(a-b)+b,g_{1}(a-b)+b,g_{1}(a-b)+b)$ ,$\varphi_{2}^{0}(a,b,a)=(g_{2}(b-a)+a,g_{2}(b-a)+a,g_{2}(b-a)+a)$ ,$\varphi_{3}^{0}(a,a,c)=(g_{3}(c-a)+a,g_{3}(c-a)+a,g_{3}(c-a)+a)$.

Thinking about the composition rule(5.10), we have$\varphi_{\infty}^{0}=\varphi_{1}^{0}\circ\varphi_{\infty}^{1}=\varphi_{2}^{0}\circ\varphi_{\infty}^{2}=\varphi_{3}^{0}\circ\varphi_{\infty}^{3}$$g_{1}(a-f_{1}(b-c)-c)+f_{1}(b-c)+c$$=g_{2}(b-f_{2}(c-a)-a)+f_{2}(c-a)+a$$=g_{3}(c-f_{3}(a-b)-b)+f_{3}(a-b)+b$ .

Grothendieck Topologies on

5.2(5.11)$\chi$

Any Grothendieck topology on we are discussing in the following has at least one sheaf forit. Therefore, we can assume any sieve on satisfies$\chi$$I$$I=\mathcal{U}.$$\mathcal{U}$Proposition 5.1. Let 1 be a Grothendieck topology on . Then,$\chi$(5.12)$1(\mathcal{U}_{k})=\{\downarrow \mathcal{U}_{k}\}$for $k=0,1,2$ or 3.
So, we only discuss about$1(\mathcal{U}_{\infty})$

below.

For $k=0,1,2,3,$ , define sieves$\infty$$\mathcal{U}_{\infty}.$$I_{12}$$:=I_{1}\cup I_{2},$$I_{13}$$,$$:=I_{1}\cup I_{3},$$I_{23}$$,$$I_{k}$

on$\mathcal{U}_{k}$by$:=I_{2}\cup I_{3}$$I_{k}$$:=\downarrow \mathcal{U}_{k}$, and$I_{123}$$:=I_{1}\cup I_{2}\cup I_{3}..$

Now, we define two Grothendieck Topologies 10 and.

.

. Followings are all possible sieves on$J_{1}.$$J_{0}$

is defined by$J_{0}(\mathcal{U}_{k})=\{I_{k}\}$for $k=0,1,2,3$ or$J_{1}$

is defined by$J_{1}(\mathcal{U}_{k})=\{I_{k}\}$for $k=0,1,2$ or 3 and$\infty.$$]_{1}(\mathcal{U}_{\infty})=\{I_{\infty}, I_{123}\}.$

We can easily show that any Grothendieck topology on that has at least one sheaf onother than 10 contains . In other words, is the smallest Grothendieck topology on next to$\chi$$J_{1}$$1_{0}.$$J_{1}$$\chi$$\chi$

178
The following diagram shows the unique extension from

$I_{123}$to$I_{\infty}.$(5.13)

So, we have a necessary and sufficient condition for a monetary value measure to be a

Proposition 5.2.

$\varphi$

becomes a sheaffor$J_{1}$

ifffor all $a,a’,b,$$J_{1}$-sheaf.

$b’,c,c’\in \mathbb{R},$$g_{1}(a-c’)+c’=g_{2}(b-a’)+a’=g_{3}(c-b’)+b’$$\Rightarrow(c’=f_{1}(b-c)+c)\wedge(a’=f_{2}(c-a)+a)\wedge(b’=f_{3}(a-b)+b)$

Entropic Value Measures on

5.3

be a probability measure on
measure defined byLet$\mathbb{P}$$\Omega$$\varphi_{\infty}^{1}$(5.14)$\chi$

defined by$\mathbb{P}=(p_{1}, p_{2}, p_{3})$$\varphi_{\mathcal{U}}^{\mathcal{V}}(X):=\frac{1}{\lambda}\log \mathbb{E}^{\mathbb{P}}[e^{\lambda X}|\mathcal{V}]$Then the function.

and$\varphi$be an entropic value.

(5.15)in (5.4) is$\varphi_{\infty}^{1}(a,b,c)=\frac{1}{\lambda}\log \mathbb{E}^{\mathbb{P}}[(e^{\lambda a},e^{\lambda b},e^{\lambda c})|\mathcal{U}_{1}]$.

$=(a, \frac{1}{\lambda}\log\frac{p_{2}e^{\lambda b}+p_{3}e^{\lambda c}}{p_{2}+p_{3}}, \frac{1}{\lambda}\log\frac{p_{2}e^{\lambda b}+p_{3}e^{\lambda c}}{p_{2}+p_{3}})$(5.16)

Therefore, the corresponding six functions defined in (5.5) and (5.9) are$f_{1}(x)= \frac{1}{\lambda}\log\frac{p_{2}e^{\lambda x}+p_{3}}{p_{2}+p_{3}},$$f_{2}(x)= \frac{1}{\lambda}\log\frac{p_{3}e^{\lambda x}+p_{1}}{p_{3}+p_{1}},$$f_{3}(x)= \frac{1}{\lambda}\log\frac{p_{1}e^{\lambda x}+p_{2}}{p_{1}+p_{2}},$$g_{1}(x)= \frac{1}{\lambda}\log(p_{1}e^{\lambda x}+p_{2}+p_{3})$,,$g_{2}(x)= \frac{1}{\lambda}\log(p_{1}+p_{2}e^{\lambda x}+p_{3})$$g_{3}(x)= \frac{1}{\lambda}\log(p_{1}+p_{2}+p_{3}e^{\lambda x})$.

So, the question is if the entropic value measure is a h-sheaf. By Proposition 5.2, its neces-sary and sufficient condition becomes like the following:$p_{1}e^{\lambda a}+(1-p_{1})e^{\lambda c’}=p_{2}e^{\lambda b}+(1-p_{2})e^{\lambda a’}=p_{3}e^{\lambda c}+(1-p_{3})e^{\lambda b’}=:Z$$\Rightarrow Z=p_{1}e^{\lambda a}+p_{2}e^{\lambda b}+p_{3}e^{\lambda c}.$However, this does not hold in general.

179
Theorem 5.3.

$\varphi$

is not a$J_{1}$-sheaf.

Corollary 5.4. Any set ofaxioms over $\Omega=\{1,2,3\}$ thatnot complete.

$acc\varphi ts$

concave monetary value measures is

6 Conclusion
We specified a concept of monetary value measures through the language of category theory
It is defined as an appropriate class of presheaves over a set of -fields as a poset. The resulting monetary value measures satisfy naturally so-called time consistency condition as well as
dynamic programming principle.
Next, we showed a concrete shape of the largest Grothendieck topology for which monetary
value measures satisfying given axioms become sheaves. By using sheafification functors, forany monetary value measure, we constructed its best approximation of the monetary value
measure that satisfies given axioms in case the axioms are complete.
As a list of future’s investigation, we will try to formulate a robust representation of concave
monetary value measures in a category-theoretic language. We also seek the possibility to represent each individual axiom of monetary value measures as a specific Grothendieck topology
which may give us an insight about different aspects of the axioms of monetary value measures. We investigate the completeness condition of sets of axioms for more realistic in orderto make sheafification functors work better.
$\sigma$$\Omega$

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Matthew Jackson, A sheaf theoretic approach to measure theory, Ph.D. thesis, Universityof Pittsburgh, 2006.

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Shigeo Kusuoka and Yuji Morimoto, Homogeneous law-invariant coherent multiperiod
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Graduate School of Intemational Corporate Strategy
Hitotsubashi University
2-1-2 Hitotsubashi, Chiyoda-ku, Tokyo, Japan-mail address: taka.adachi@gmail.com$E$$-\ovalbox{\tt\small REJECT}*\not\cong x\not\cong\beta_{\pi\Phi \mathbb{P}_{T\hat{JE_{-}}\ovalbox{\tt\small REJECT}\otimes\#@ffl_{J^{2}\iota}^{pc}EE^{rightarrow}\theta}}^{dr}\underline{|l}_{\overline{|\Leftrightarrow}_{u}’}$