Monetary Value Measure as a Contravariant Functor (Category Theory) Part 1

category_concepts

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 ρ = LP (Ω, F, P) → ℜ satisfying the following axioms

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 = LP (Ω, F, P) is the space of equivalence classes of ℜ-valued random variables which are bounded by the || . ||P form.

2.0 For a σ-field U ⊂ F, L(U) = L (Ω, U, P | U) is the space of all equivalence classes of bounded ℜ-valued random variables, equipped with the usual sup form.

3.0 Let F = {Ft}t∈[0,T] be a filtration. A dynamic monetary value measure is a collection of functions ψ = {Ψt : L(FT) → L(Ft)}t∈[0,T] satisfying

3.1 Cash Invariance (∀ X ∈ L(FT))(∀ Z ∈ L(FT)) Ψt (X + Z) = Ψt (X) + Z

3.2 Monotonicity (∀ X ∈ L(FT))(∀ Y ∈ L(FT)) X ≤ Y ⇒ Ψt (X) ≤ Ψt (Y)

3.3 Normalization Ψt (0) = 0

Note 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(FT)) Ψs (X) =  Ψst (X))

Axiom 5.0 Time Consistency: For 0 ≤ s ≤ t ≤ T, (∀ X, ∀ Y ∈  L(FT)) Ψt (X) ≤ Ψt (Y) ⇒ Ψs (X) ≤ Ψs (Y)

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 OC of objects and a collection of MC of arrows or morphisms such that

6.0 There are two functions MCdom OC & MC →cod OC

When dom(f) = A and cod (f) = B, we write f : A → B

We define a so-called hom-set of given objects A and B by

HomC(A, B) := {f ∈ MC | f : A → B}

6.1 For f : A → B & g : B → C, there is an arrow g o f : A → C called the composition of g and f. 

6.2 Every object A is associated with an identity arrow 1A : A → A, such that f o 1A = f and 1A o g = g where dom(f) = A & cod(g) = g

7.0 Functors: Let C and D be two categories. A functor F: C → D consists of two functions

FO : OC → OD and FM : MC → MD

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(1A) = 1F(A)

8.0 Contravariant Functors: A functor F : Cop → D is called a contravariant functor if 7.1 and 7.2 are replaced by

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 C →F D and C →G D be two functors. A natural transformation α : F →. G consists of a family of arrows 〈αC | C ∈ OCmaking the following diagram commute

C1          F(C1) —>αC1 G(C1)

f↓       F(f) ↓             G(f)↓

C2         F(c2) —>αC2 G(C2)

10.0 Functor Categories: Let C and D be categories. A functor category DC is the category such that

10.1 ODC := collection of all functors from C to D

10.2 HomDC (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 χ := χ(F) be the et of all sub-fields of F. It becomes a poset with the set inclusion relation χ becomes a category whose hom set Homχ(V, U) for U, V ∈ χ is defined by

Homχ(V, U) := iVU if V ⊂ U

:= Φ otherwise.

The arrow iVU 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 : χop → Set

satisfying the following two conditions

13.1 for U ∈ χ, Ψ(U) := L(U)

13.2 for U, V ∈ χ, such that V ⊂ U, the map ΨVU := Ψ(iVU) : L(U) → L(V) satisfies

13.3 cash invariance: (∀ X ∈ L(U))(∀ Z ∈ L(V)) ΨVU (X + Z) =  ΨVU (X) + Z

13.4 monotonicity: (∀ X ∈ L(U)) (∀ Y ∈ L(U)) X ≤ Y ⇒ ΨVU(X) ≤ ΨVU(Y)

13.5 normalization: ΨVU(0) = 0

At 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 Ψ is a contravariant functor, and thus for any triple σ-fields W ⊂ V ⊂ U in χ, we have

13.6 ΨUU = 1L(U) and ΨWV o ΨVU = ΨWU

14.0 Concave monetary value measure: A monetary value measure Ψ is said to be concave if for any V ⊂ U in χ, X, Y ∈ L(U) and λ ∈ [0,1],

ΨVU(λX + (1- λ)Y) ≥ λΨVU(X) + (1-λ)ΨVU(Y)

An entropic value measure is concave.

Here are some properties of monetary value measures.

15.0 Proposition: Let Ψ : χop → Set be a monetary value measure, and W ⊂ V ⊂ U be σ-fields in χ.

15.1 (∀ X ∈ L(V)) ΨVU(X) = X

By cash invariance and normalization, ΨVU(X) = ΨVU(0 + X) = ΨVU(0) + X = X

15.2 Idempotentness: (∀ X ∈ L(U)) ΨVUΨVU(X) = ΨVU(X)

Since, ΨVU(X) ∈  L(V), it is obvious by 15.1

15.3 Local property: (∀ X ∈ L(U))(∀ Y ∈ L(U))(∀ A ∈  V) ΨVU (1AX + 1ACY) = 1AΨVU(X) + 1AC ΨVU(Y) 

First we show for any A ∈ V,

1AΨVU(X) = 1AΨVU(1AX)

Since, X ∈ L(Ω, U, P), we have |X| ≤ ||X||

therefore,

1AX – 1AC||X|| ≤ 1AX + 1ACX ≤ 1AX + 1AC||x||

hence, by cash invariance and monotonicity,

ΨVU(1AX) – 1AC||x|| = ΨVU(1AX – 1AC||x||) ≤ ΨVU(X) ≤ ΨVU(1AX) + 1AC||x||)

then,

1AΨVU(1AX) = 1AVU(1AX) – 1AC||x||) ≤ 1AΨVU(X) ≤ 1AVU(1AX) + 1AC||x||) = 1AVU(1AX)

getting 15.3

Using 15.3 twice, we have

ΨVU (1AX + 1ACY) = 1AΨVU(1AX + 1ACY) + 1ACΨVU(1AX + 1ACY)

1AΨVU(1A(1AX + 1ACY)) + 1ACΨVU(1AX + 1ACY))

1AΨVU(1AX) + 1ACΨVU(1ACY)

1AΨVU(X) + 1ACΨVU(Y)

15.4 Dynamic programming principle: (∀ X ∈ L(U)) ΨWU(X) = ΨWUVU(X))

by way of dynamic risk measure and monetary value measure,

ΨWU(X) = ΨWVVU(X)) =  ΨWVVUVU(X))) = (ΨWV o ΨVU)(ΨVU(X)) = ΨWUVU(X))

15.5 Time consistency: (∀ X ∈ L(U))(∀ Y ∈ L(U)) (ΨVU(X)) ≤ (ΨVU(Y)) ⇒ ΨWU(X) ≤ ΨWU(Y)

Assuming ΨVU(X) ≤ ΨVU(Y), then, by monotonicity and monetary value measure,

ΨWU(X) = ΨWVVU(X)) ≤ ΨWVVU(Y)) = ΨWU(Y)

……………

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