Entropic Value Measurement and Monetary Value Measures. End Part.

Any Grothendieck topology on χ we are discussing in the following has at least one sheaf for it. Therefore, we can assume any sieve I on U satisfying ∨ I =U.

Let J be a Grothendieck topology on χ. Then,

J{U_K} = {↓ U_K}

for K = 0, 1, 2, or 3

Discussing about J(U_∞) 

for k = 0, 1, 2, 3, ∞, define sieves I_K on U_K by I_K := ↓ U_K

Followings are all possible sieves on U_∞.

I₁₂ = I₁ ∪ I₂, I₁₃ = I₁ ∪ I₃, I₂₃ = I₂ ∪ I₃, I₁₂₃ = I₁ ∪ I₂ ∪ I₃

Now, we define two Grothendieck topologies J₀ and J₁

J₀ is defined as J₀(U_K) = {I_K}, for K = 0, 1, 2, 3, or ∞

J₁ is defined as J₁(U_K) = {I_K}, for K = 0, 1, 2, 3 or J₁(U_∞) = {I_∞,  I₁₂₃}

We can easily show that any Grothendieck topology on χ that has at least one sheaf on χ other than J₀ contains J₁. In other words, J₁ is the smallest Grothendieck topology on χ next to J₀.

The diagram shows the unique extension from I₁₂₃ to I_∞

So, we have a necessary and sufficient condition for a monetary value measure to be a J_1-sheaf.

1.0 Ψ becomes a sheaf for J₁ iff ∀ a, a’, b, b’, c, c’ ∈ ℜ

g₁ (a – c’) + c’ = g₂ (b – a’) + a’= g₃ (c – b’) + b’

⇒ (c’ = f₁ (b – c) + c) ∧ (a’ = (f₂ (c – a) + a) ∧ (b’ = f₃ (a – b) + b)

Entropic value measurement: Let P be a probability measure on Ω defined by P = (p₁, p₂, p₃) and Ψ be the entropic value measured by

Ψ^V_U(X) := 1/λ log E^P [e^λX | V]

Then from

Ψ¹_∞ (a, b, c)  = 1/λ log E^P [(e^λa, e^λb, e^λc) | U₁]

= (a, 1/λ log (p₂e^λb + p₃e^λc)/(p₂ + p₃), 1/λ 1/λ log (p₂e^λb + p₃e^λc)/(p₂ + p₃))

the corresponding six functions from part 3 now are,

f₁(x) = 1/λ log (p₂e^λx + p₃)/(p₂ + p₃)

f₂(x) = 1/λ log (p₃e^λx + p₁)/(p₃ + p₁)

f₃(x) = 1/λ log (p₁e^λx + p₂)/(p₁ +p₂)

g₁(x) = 1/λ log (p₁e^λx + p₂ + p₃)

g₂(x) = 1/λ log (p₁ + p₂e^λx + p₃)

g₃(x) = 1/λ log (p₁ + p₂ + p₃e^λx)

So, the question is if the entropic value measure is a J_1-sheaf. The necessary and sufficient condition becomes like

p₁e^λa + (1 – p₁)e^λc’ = p₂e^λb + (1 – p₂)e^λa’ = p₃e^λc + (1 – p₃)e^λb’ := Z

⇒ Z = p₁e^λa + p₂e^λb + p₃e^λc

However, this does not hold true in general. On the corollary, any set of axioms on Ω = {1, 2, 3} that accepts concave monetary value measures is not complete.

The concept of monetary value measures through the language of category theory is defined as an appropriate class of presheaves over a set of σ-fields as a poset. The resulting monetary value measures satisfy naturally so-called time consistency condition as well as dynamic programming principle. Next, we saw how a concrete shape of the largest Grothendieck topology for which monetary value measures satisfying given axioms become sheaves. By using sheafification functors, for any monetary value measure, it is possible to construct its best approximation of the monetary value measure that satisfies given axioms in case the axioms are complete.

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