Sheafification Functor and Arbitrary Monetary Value Measure. Part 3

Here are Part 1 and Part 2

Let A be a fixed set of axioms. Then for a given arbitrary monetary value measure Ψ can we make a good alternative for it? In other words, can we find a monetary value measure that satisfies A and is the best approximation of the original Ψ? For a Grothendieck topology J on χ, define Sh(χ, J) ⊂ Setχop to be a full sub-category whose objects are all sheaves for J. Then, it is well known that  a left adjoint πJ in the following diagram.

Sh(χ, J) → Setχop

Sh(χ, J) ←πJ Setχop

πJ (Ψ) ← Ψ

The functor πJ is known as Sheafification functor, which has the following limit cone:

for sieves I, K and U. This also satisfies the following theorem.

1.0 If πJ (Ψ) is a sheaf for J

1.1 If Ψ is a sheaf for J, then for any U ∈ χ,  πJ (Ψ)(U) ≅ L(U)

The theorem 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.

2.0 Let A be a set of axioms of monetary value measures

2.1 M(A) := the collection of all monetary value measures satisfying A

2.2 MO := collection of all monetary value measures

2.3 A is called complete if

πJM(A) (MO) ⊂ M(A)

3.0 Let A be a complete set of axioms. Then, for a monetary value measure Ψ ∈ MO, πJM(A(Ψ) is the monetary value measure that is the best approximation satisfying A.

Let us investigate if the set of axioms of concave monetary value measures is complete in the case of Ω = {1, 2, 3} with a σ-field F := 2Ω

We enumerate all possible sub-σ-fields of Ω, that is, the shape of the category χ = χ(Ω),

where,

U := F := 2Ω

U1 := {Φ, {1}, {2, 3}, Ω}

U2 := {Φ, {2}, {1, 3}, Ω}

U3 := {Φ, {3}, {1, 2}, Ω}

U4 := {Φ, Ω}

The Banach spaces defined by the elements of χ are

L := L := L(U) := {a, b, c | a, b, c ∈ ℜ}

L1 := L(U1) := {a, b, b | a, b ∈ ℜ}

L2 := L(U2) := {a, b, a | a, b ∈ ℜ}

L3 := L(U3) := {a, a, c | a, c ∈ ℜ}

L0 := L(U0) := {a, a, a, | a ∈ ℜ}

Then a monetary value measure Ψ : χop → Set on χ is determined by the following six functions

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

For Ψ1 : L → L1, we have by the cash invariance axiom,

Ψ1 (a, b, c) = Ψ1 ((0, b – c, 0) + (a, c, c))

= Ψ1 ((0, b – c, 0)) + (a, c, c)

= (f12 (b – c), f11 (b – c), f11 (b – c)) + (a, c, c)

= (f12 (b – c) + a, f11 (b – c) + c, f11 (b – c)+ c)

where f11, f12 : ℜ → ℜ are defined by (f12(x), f11(x), f11(x)) = Ψ1 (0, x, 0).

Similarly, if we define nine functions

f11, f12, f21, f22, f31, f32, g1, g2, g3ℜ → ℜ by

(f12(x), f11(x), f11(x)) = Ψ1(0, x, 0)

(f21(x), f22(x), f21(x)) = Ψ2(0, 0, x)

(f31(x), f31(x), f32(x)) = Ψ3(x, 0, 0)

(g1(x), g1(x), g1(x)) = Ψ01(x, 0, 0)

(g2(x), g2(x), g2(x)) = Ψ02(0, x, 0)

(g3(x), g3(x), g3(x)) = Ψ03(0, 0, x)

We can represent the original six functions by nine functions

Ψ1(a, b, c) = (f12(b – c) + a, f11(b – c) + c, f11(b – c) + c),

Ψ2(a, b, c) = (f21(c – a) + a, f22(c – a) + a, f21(c – a) + a),

Ψ3(a, b, c) = (f31(a – b) + b, f31(a – b) + b, f32(a – b) + c),

Ψ01(a, b, b) = (g1(a – b) + b, g1(a – b) + b, g1(a – b) + b),

Ψ02(a, b, a) = (g2(b – a) + a, g2(b – a) + a, g2(b – a) + a),

Ψ03(a, a, c) = (g3(c – a) + a, g3(c – a) + a, g3(c – a) + a)

Next by the normahzation axiom, we have

f11(0) = f12(0) = f21(0) = f22(0) = f31(0) = f32(0) = g1(0) = g2(0) = g3(0) = 0

partially differentiating the function in Ψ1(a, b, c), we have

Ψ1(a, b, c)/∂a = (1, 0, 0)

∂Ψ1(a, b, c)/∂b = (f’12(b – c), f’11(b – c), f’11(b – c))

Ψ1(a, b, c)/∂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 ≤ f’11 ≤ 1. Then by the result of the normalization axiom, we have

x ∈ ℜ, f12(x) = 0. Hence, ∀ x ∈ ℜ,

f12(x) = f22(x) = f32(x) = 0

With this knowledge, let us redefine the three functions f1, f2, f3 : ℜ → ℜ by

(0, f1(x), f1(x)) = Ψ1(0, x, 0)

(f2(x), 0, f2(x)) = Ψ2(0, 0, x)

(f3(x), f3(x), 0) = Ψ3(x, 0, 0)

Then, we have a new representation of the original six functions

Ψ1(a, b, c) = (a, f1(b – c) + c, f1(b – c) + c)

Ψ2(a, b, c) = (f2(c – a) + a, b, f2(c – a) + a)

Ψ3(a, b, c) = (f3(a – b) + b, f3(a – b) + b, c)

Ψ01(a, b, b) = (g1(a – b) + b, g1(a – b) + b, g1(a – b) + b)

Ψ02(a, b, a) = (g2(b – a) + a, g2(b – a) + a, g2(b – a) + a)

Ψ03(a, a, c) = (g3(c – a) + a, g3(c – a) + a, g3(c – a) + a)

Thinking about the composition rule, we have

Ψ0 = Ψ01 o Ψ1 = Ψ02 o Ψ2 = Ψ03 o Ψ3

g1(a – f1(b – c) – c) + f1(b – c) + c

= g2(b – f2(c – a) – a) + f2(c – a) + a

=g3(c – f3(a – b) – b) + f3(a – b) + b

………..

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

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)

……………