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This paper is devoted to the connection between the probability distributions which produce solutions of the one-dimensional, time-independent Schr?dinger Equation and the Risk Measures’ Theory. We deduce that the Pareto, the Generalized Pareto Distributions and in general the distributions whose support is a pure subset of the positive real numbers, are adequate for the definition of the so-called Quantum Risk Measures. Thanks both to the finite values of them and the relation of these distributions to the Extreme Value Theory, these new Risk Measures may be useful in cases where a discrimination of types of insurance contracts and the volume of contracts has to be known. In the case of use of the Quantum Theory, the mass of the quantum particle represents either the volume of trading in a financial asset, or the number of insurance contracts of a certain type.

As it is mentioned in [

Definition 1.1. An operator A on a Hilbert space

holds.

Definition 1.2. An operator

Definition 1.3. A Weyl sequence for the operator A and the eigenvalue

Definition 1.4. The continuous spectrum of an operator A is the set of the eigenvalues

Definition 1.5. The point spectrum of an operator A, where

Definition 1.6. (Weyl’s Theorem) If

In the case of the one-dimensional quantum models

for any wave function

The elementaries of quantum finance denote that any asset is a quantum particle, whose changes in position x correspond to the changes of its value. The changes of its value are decomposed into the kinetic energy of the particle, which is the total effort of the investors to change its value. For this reason, the mass m of this particle, denotes the total number of investors which are involved into investements to this asset. On the other hand, the dynamic energy denotes the changes of the value whose cause is some exogenous factor being a function of the position x of the quantum particle. This is the meaning of the function of the potential

of the Hamiltonian’s Spectrum, denotes that the set of the possible Asset Monetary Values is the set of the Hamiltonian’s Spectrum (

which denotes that any wave-function is a squarely-integrable function. By the term wave-function we mean any solution of the above time-independent Schrödinger Equation. Every wave-function

In this paper we prove an essential Theorem on what it may be called Quantum Risk Theory. This Theorem refers to any family

the form

then:

1) If

the wave-function

2) The Spectrum of the Hamiltonian

3) The associated Coherent Risk Measure

4) The brackets

We also give specific Examples of classes

Under the above frame, for a Hamiltonian H associated with a continuous spectrum, or else the set of the eigenvalues of H contain an open set of

where

The above theorem is essential:

Theorem 3.1. If S(H) contains an open set of R, then the quantum risk measure

We remind that the Hermitian identity operator

Proof. We verify the four properties of coherence.

1)

2) Two assets coincide with two Hamiltonians

3) The Positive Homogeneity arises from the fact that for any specific

4) Finally, the Monotonicity arises from the fact that if for two Hamiltonians

holds for any

Theorem 4.1. Consider a family

family

then:

1) If

the wave-function

2) The Spectrum of the Hamiltonian

3) The associated Coherent Risk Measure

4) The brackets

Proof. 1) For the function

2)

3)

4)

Hence, if

Example 4.2. The Pareto Family of Distributions

Also if

Example 4.3. The Generalized Pareto Family of Distributions

the form

The conclusion of the paper is that the notion of risk measure may be extended in a quantum finance framework, as far as it may be applied on a time-independent Hamiltonian operator and specifically on its continuous spec- trum. The value of such a risk measure is finite and in the case of Pareto and Generalized Pareto distributions is negative. This risk model may be applied either in the case of reinsurance pricing, or in the case where no other known model is developed like naval insurance contracts.

Christos E.Kountzakis,Maria P.Koutsouraki, (2016) On Quantum Risk Modelling. Journal of Mathematical Finance,06,43-47. doi: 10.4236/jmf.2016.61005