Let us get a very brief review of dynamic risk measure theory and category theory.

* 1.0* A one period monetary risk measure is a function

*satisfying the following axioms*

**ρ = L**^{P}(Ω, F, P) → ℜ* 1.1* Cash Invariance

**(∀X) (∀ a ∈ ℜ) ρ (X + a) = ρ (X) – a*** 1.2* Monotonicity

**(∀ X) (∀ Y) X ≤ Y ⇒ ρ (X) ≥ ρ (Y)*** 1.3* Normalization

**ρ (0) = 0**where * ρ = L^{P} (Ω, F, P)* is the space of equivalence classes of

*which are bounded by the*

**ℜ-valued random variables***form.*

**|| . ||**_{P}* 2.0* For a

*is the space of all equivalence classes of bounded*

**σ-field U ⊂ F, L(U) = L**^{∞}(Ω, U, P | U)*, equipped with the usual sup form.*

**ℜ-valued random variables*** 3.0* Let

*be a filtration. A dynamic monetary value measure is a collection of functions*

**F = {F**_{t}}_{t∈[0,T]}*satisfying*

**ψ = {Ψ**_{t}: L(F_{T}) → L(F_{t})}_{t∈[0,T]}* 3.1* Cash Invariance

**(∀ X ∈****L(F**_{T}))(∀ Z ∈ L(F_{T})) Ψ_{t}(X + Z) = Ψ_{t}(X) + Z* 3.2* Monotonicity

**(∀ X ∈****L(F**_{T}))(∀ Y ∈ L(F_{T})) X ≤ Y ⇒ Ψ_{t}(X) ≤ Ψ_{t}(Y)* 3.3 *Normalization

**Ψ**_{t}(0) = 0Note that the directions of some inequalities in Definition 1.0-1.3 are different from those of Definition 3.0-3.3, because we now are dealing with 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.

* Axiom 4.0* Dynamic Programming Principle: For

**0 ≤ s ≤ t ≤ T,****(∀ X ∈****L(F**_{T})) Ψ_{s}(X) =

**Ψ**_{s}(Ψ_{t}(X))* Axiom 5.0* Time Consistency: For

**0 ≤ s ≤ t ≤ T,****(∀ X, ∀ Y ∈ L(F**_{T})) Ψ_{t}(X) ≤ Ψ_{t}(Y) ⇒ Ψ_{s}(X) ≤*(Y)***Ψ**_{s}Category theory is an area of study in mathematics that examines in an abstract way the properties of maps (called morphisms or arrows) satisfying some basic conditions.

A Category * C* consists of a collection of

*of objects and a collection of*

**O**_{C}*of arrows or morphisms such that*

**M**_{C}* 6.0* There are two functions

*&*

**M**_{C}→^{dom}O_{C}

**M**_{C}→^{cod}O_{C}When * dom(f) = A* and

*, we write*

**cod (f) = B**

**f : A → B**We define a so-called * hom-set* of given objects A and B by

**Hom _{C}(A, B) := {f ∈ MC | f : A → B}**

* 6.1* For

*&*

**f : A → B***there is an arrow*

**g : B → C,**

**called the composition of**

*g o f : A***→ C***and*

**g***.*

**f**

* 6.2* Every object A is associated with an identity arrow

*, such that*

**1**_{A}: A → A*and*

**f o 1**_{A}= f*where*

**1**_{A}o g = g

**dom(f) = A & cod(g) = g*** 7.0* Functors: Let

*and*

**C***be two categories. A functor*

**D***consists of two functions*

**F: C → D*** F_{O} : O_{C} → O_{D} *and

**F**_{M}: M_{C}→ M_{D}satisfying

**7.1 f : A → B ⇒ F(f) : F(A) → F(B)**

**7.2 F(g o f) = F(g) o F(f)**

**7.3 F(1 _{A}) = 1_{F(A)}**

* 8.0* Contravariant Functors: A functor

*is called a contravariant functor if*

**F : C**^{op}→ D*and*

**7.1***are replaced by*

**7.2****8.1 f : A → B ⇒ F(f) : F(B) → F(A)**

**8.2 F(g o f) = F(f) o F(g)**

* 9.0* Natural Transformations: Let

*and*

**C →**^{F}D*be two functors. A natural transformation*

**C →**^{G}D*consists of a family of arrows*

**α : F →**^{.}G*making the following diagram commute*

**〈α**_{C}| C ∈ O_{C}〉**C _{1} F(C_{1}) —>^{αC1} G(C_{1})**

**f↓ F(f) ↓ G(f)↓**

**C _{2} F(c_{2}) —>^{αC2} G(C_{2})**

* 10.0* Functor Categories: Let

*and*

**C***be categories. A functor category*

**D***is the category such that*

**D**^{C}* 10.1 O_{DC}* := collection of all functors from C to D

* 10.2 Hom_{DC} (F, G)* := collection of all natural transformations from F to G.

Now, for defining monetary value measures with the language of category theory, we introduce a simple category that is actually a partially-ordered set derived by the * σ-field F*.

* 11.0* Let

*be the et of all sub-fields of*

**χ := χ(F)***. It becomes a poset with the set inclusion relation*

**F***.*

**⊂***becomes a category whose hom set*

**χ***for*

**Hom**_{χ}(V, U)*is defined by*

**U, V ∈ χ****Hom _{χ}(V, U) := i^{V}_{U} if V ⊂ U**

* := Φ* otherwise.

The arrow **i ^{V}_{U} is called the inclusion map. **

* 12.0 ⊥ := {Ω, Φ}*, which is the least element of

**χ.*** 13.0* Monetary Value Measure is a contravariant functor

**Ψ : χ ^{op} → Set**

satisfying the following two conditions

* 13.1* for

**U ∈ χ, Ψ(U) := L(U)*** 13.2* for

*, such that*

**U, V ∈ χ***, the map*

**V ⊂ U***satisfies*

**Ψ**^{V}_{U}:= Ψ(i^{V}_{U}) : L(U) → L(V)* 13.3* cash invariance:

**(∀ X ∈ L(U))(∀ Z ∈ L(V))**

**Ψ**^{V}_{U}(X + Z) = Ψ^{V}_{U}(X) + Z* 13.4* monotonicity:

**(∀ X ∈ L(U)) (∀ Y ∈ L(U)) X ≤ Y ⇒ Ψ**^{V}_{U}(X) ≤ Ψ^{V}_{U}(Y)* 13.5* normalization:

**Ψ**^{V}_{U}(0) = 0At this point, we do not require the monetary value measures to satisfy some familiar con- ditions 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 parameters from * 13.0* is that

*is a contravariant functor, and thus for any triple*

**Ψ***, we have*

**σ-fields W ⊂ V ⊂ U in χ*** 13.6 Ψ^{U}_{U} = 1_{L(U)} *and

**Ψ**^{W}_{V}o Ψ^{V}_{U}= Ψ^{W}_{U}* 14.0* Concave monetary value measure: A monetary value measure

*is said to be concave if for any*

**Ψ***and*

**V ⊂ U in χ, X, Y ∈ L(U)***,*

**λ ∈ [0,1]****Ψ ^{V}_{U}(λX + (1- λ)Y) ≥ λΨ^{V}_{U}(X) + (1-λ)Ψ^{V}_{U}(Y)**

An entropic value measure is concave.

Here are some properties of monetary value measures.

* 15.0* Proposition: Let

*be a monetary value measure, and*

**Ψ : χ**^{op}→ Set*be*

**W ⊂ V ⊂ U***in*

**σ-fields***.*

**χ****15.1** **(∀ X ∈ L(V)) Ψ ^{V}_{U}(X) = X**

By cash invariance and normalization, **Ψ ^{V}_{U}(X) = Ψ^{V}_{U}(0 + X) = Ψ^{V}_{U}(0) + X = X**

* 15.2* Idempotentness:

**(∀ X ∈ L(U))**

**Ψ**^{V}_{U}Ψ^{V}_{U}(X) = Ψ^{V}_{U}(X)Since, **Ψ ^{V}_{U}(X) **

*, it is obvious by*

**∈ L(V)**

**15.1*** 15.3* Local property:

**(∀ X ∈ L(U))**

**(∀ Y ∈ L(U))(∀ A ∈ V) Ψ**^{V}_{U}(1_{A}X + 1_{AC}Y) = 1_{A}Ψ^{V}_{U}(X) + 1_{AC}Ψ^{V}_{U}(Y)First we show for any * A ∈ V*,

**1 _{A}Ψ^{V}_{U}(X) = 1_{A}Ψ^{V}_{U}(1_{A}X)**

Since, * X ∈ L^{∞}(Ω, U, P), *we have

**|X| ≤ ||X||**_{∞}therefore,

**1 _{A}X – 1_{AC}||X||_{∞} ≤ 1_{A}X + 1_{AC}X ≤ 1_{A}X + 1_{AC}||x||_{∞}**

hence, by cash invariance and monotonicity,

**Ψ ^{V}_{U}(1_{A}X) – 1_{AC}||x||_{∞} = Ψ^{V}_{U}(1_{A}X – 1_{AC}||x||_{∞}) ≤ Ψ^{V}_{U}(X) ≤ Ψ^{V}_{U}(1_{A}X) + 1_{AC}||x||_{∞})**

then,

**1 _{A}Ψ^{V}_{U}(1_{A}X) = 1_{A}(Ψ^{V}_{U}(1_{A}X) – 1_{AC}||x||_{∞}) ≤ 1_{A}Ψ^{V}_{U}(X) ≤ 1_{A}(Ψ^{V}_{U}(1_{A}X) + 1_{AC}||x||_{∞}) = 1_{A}(Ψ^{V}_{U}(1_{A}X)**

getting **15.3**

Using * 15.3* twice, we have

**Ψ ^{V}_{U} (1_{A}X + 1_{AC}Y) = 1_{A}Ψ^{V}_{U}(1_{A}X + 1_{AC}Y) + 1_{AC}Ψ^{V}_{U}(1_{A}X + 1_{AC}Y)**

= **1 _{A}Ψ^{V}_{U}(1_{A}(1_{A}X + 1_{AC}Y)) + 1_{AC}Ψ^{V}_{U}(1_{A}X + 1_{AC}Y))**

= **1 _{A}Ψ^{V}_{U}(1_{A}X) + 1_{AC}Ψ^{V}_{U}(1_{AC}Y)**

= **1 _{A}Ψ^{V}_{U}(X) + 1_{AC}Ψ^{V}_{U}(Y)**

* 15.4* Dynamic programming principle:

**(∀ X ∈ L(U)) Ψ**^{W}_{U}(X) = Ψ^{W}_{U}(Ψ^{V}_{U}(X))by way of dynamic risk measure and monetary value measure,

**Ψ ^{W}_{U}(X) = Ψ^{W}_{V}(Ψ^{V}_{U}(X)) = Ψ^{W}_{V}(Ψ^{V}_{U}(Ψ^{V}_{U}(X))) = (Ψ^{W}_{V} o Ψ^{V}_{U})(Ψ^{V}_{U}(X)) = Ψ^{W}_{U}(Ψ^{V}_{U}(X))**

* 15.5* Time consistency:

**(∀ X ∈ L(U))**

**(∀ Y ∈ L(U)) (Ψ**^{V}_{U}(X)) ≤ (Ψ^{V}_{U}(Y)) ⇒ Ψ^{W}_{U}(X) ≤ Ψ^{W}_{U}(Y)Assuming * Ψ^{V}_{U}(X) ≤ Ψ^{V}_{U}(Y)*, then, by monotonicity and monetary value measure,

**Ψ ^{W}_{U}(X) = Ψ^{W}_{V}(Ψ^{V}_{U}(X)) ≤ Ψ^{W}_{V}(Ψ^{V}_{U}(Y)) = Ψ^{W}_{U}(Y)**

……………

[…] contravariant functor ρ : Cop → Set is called a presheaf for a category C. By definition, a monetary value measure […]

[…] are Part 1 and Part […]