# The quantum Black-Scholes equation.

Abstract

Motivated by the work of Segal and Segal in [16] on the Black-Scholes pricing formula in the quantum context, we study a quantum extension of the Black-Scholes equation within the context of Hudson-Parthasarathy quantum stochastic calculus,. Our model includes stock markets described by quantum Brownian motion and Poisson process.

Keywords: Option Pricing, Black-Scholes equation, Quantum Stochastic Calculus.

2000 Mathematics Subject Classification: 81S25, 91B70.

**********

1. The Merton-Black-Scholes Option Pricing Model

An option is a ticket which is bought at time t = 0 and which allows the buyer at (in the case of European call options) or until (in the case of American call options) time t = T (the time of maturity of the option) to buy a share of stock at a fixed exercise price K. In what follows we restrict to European call options. The question is: how much should one be willing to pay to buy such an option? Let [X.sub.T] be a reasonable price. According to the definition given by Merton, Black, and Scholes (M-B-S) an investment of this reasonable price in a mixed portfolio (i.e part is invested in stock and part in bond) at time t = 0, should allow the investor through a self-financing strategy (i.e one where the only change in the investor's wealth comes from changes of the prices of the stock and bond) to end up at time t = T with an amount of [([X.sub.T] K).sup.+]:= max(0, [X.sub.T] - K) which is the same as the payoff, had the option been purchased (cf. [12]). Moreover, such a reasonable price allows for no arbitrage i.e, it does not allow for risk free unbounded profits. We assume that there are no transaction costs and that the portfolio is not made smaller by consumption. If ([a.sub.t], [b.sub.t]), t [member of] [0, T] is a self -financing trading strategy (i.e an amount [a.sub.t] is invested in stock at time t and an amount [b.sub.t] is invested in bond at the same time) then the value of the portfolio at time t is given by [V.sub.t] = [a.sub.t] [X.sub.T]+ [b.sub.t] [[beta].sub.t] where, by the self-financing assumption, d[V.sub.t] = [a.sub.t] d[X.sub.T] + [b.sub.t] d[[beta].sub.t]. Here [X.sub.T] and [[beta].sub.t] denote, respectively, the price of the stock and bond at time t. We assume that d[X.sub.T] = c [X.sub.T] [d.sub.t] + [sigma] [X.sub.T] d[B.sub.t] and d[[beta].sub.t] = [[beta].sub.t] r dt where [B.sub.t] is classical Brownian motion, r > 0 is the constant interest rate of the bond, c > 0 is the mean rate of return, and [sigma] > 0 is the volatility of the stock. The assets [a.sub.t] and [b.sub.t] are in general stochastic processes. Letting [V.sub.t] = u(T - t, [X.sub.T]) where [V.sub.T] = u(0, [X.sub.T]) = [([X.sub.T] - K).sup.+] it can be shown (cf. [12]) that u(t, x) is the solution of the Black-Scholes equation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

and it is explicitly given by

u(t, x) = x [PHI](g(t, x)) - K [e.sup.-rt] [PHI](h(t, x))

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

Thus a reasonable (in the sense described above) price for a European call option is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

and the self-financing strategy ([a.sub.t], [b.sub.t]), t [member of] [0, T] is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

2. Quantum Extension of the M-B-S Model

In recent years the fields of Quantum Economics and Quantum Finance have appeared in order to interpret erratic stock market behavior with the use of quantum mechanical concepts (cf. [3], [4], [6]-[9], [11], and [14]-[16]). While no approach has yet been proved prevalent, in [16] Segal and Segal introduced quantum effects into the Merton-Black-Scholes model in order to incorporate market features such as the impossibility of simultaneous measurement of prices and their instantaneous derivatives. They did that by adding to the Brownian motion [B.sub.t] used to represent the evolution of public information affecting the market, a process [Y.sub.t] which represents the influence of factors not simultaneously measurable with those involved in [B.sub.t]. They then sketched a calculus for dealing with such processes. Segal and Segal concluded that the combined process a [B.sub.t] + b [Y.sub.t] may be represented as (in their notation) [PHI] ((a + ib) [[chi].sub.[0,t]]) where for a Hilbert space element f, [e.sup.i[PHI](f)] is the corresponding Weyl operator, and [[chi].sub.[0,t]] is the characteristic function of the interval [0, t]. In the context of the Hudson-Parthasarathy quantum stochastic calculus of [10] and [13] (see Theorem 20.10 of [13]) simple linear combinations of [PHI](f) and [PHI](i f) define the Boson Fock space annihilator and creator operators [A.sub.f] and [A.sup.[dagger].sub.f]. Segal and Segal used [PHI]([[chi].sub.[0,t]]) as the basic integrator process with integrands restricted to a special class of exponential processes. In view of the above reduction of [PHI] to A and [A.sup.[dagger]], it makes sense to study option pricing using as integrators the annihilator and creator processes of Hudson-Parthasarathy quantum stochastic calculus, thus exploiting its much larger class of integrable processes than the one considered in [16]. The Hudson-Parthasarathy calculus has a wide range of applications. For applications to, for example, control theory we refer to [2], [5] and the references therein. Quantum stochastic calculus was designed to describe the dynamics of quantum processes and we propose that we use it to study the non commutative Merton-Black-Scholes model in the following formulation (notice that our model includes also the Poisson process): We replace (see [1] for details on quantization) the stock process {[X.sub.T] / t [greater than or equal to] 0} of the classical Black-Scholes theory by the quantum mechanical process [j.sub.t](X) = [U.sup.*.sub.t] X [cross product] 1 [U.sub.t] where, for each t [greater than or equal to] 0, [U.sub.t] is a unitary operator defined on the tensor product H [cross product] [GAMMA]([L.sup.2]([R.sub.+], C)) of a system Hilbert space H and the noise Boson Fock space [GAMMA] = [GAMMA]([L.sup.2]([R.sub.+], C)) satisfying

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (0.1)

where X > 0, H, L, S are in B(H), the space of bounded linear operators on H, with S unitary and X, H self-adjoint. We identify time-independent, bounded, system space operators x with their ampliation x [cross product] 1 to H [cross product] [GAMMA]([L.sup.2]([R.sub.+], C)). The value process [V.sub.t] is defined for t [member of] [0, T] by [V.sub.t] = [a.sub.t] [j.sub.t](X) + [b.sub.t] [[beta].sub.t] with terminal condition [V.sub.T] = [([j.sub.T](X) - K).sup.+] = max(0, [j.sub.T] (X) - K), where K > 0 is a bounded self-adjoint system operator corresponding to the strike price of the quantum option, [a.sub.t] is a real-valued function, [b.sub.t] is in general an observable quantum stochastic processes (i.e [b.sub.t] is a self-adjoint operator for each t [greater than or equal to] 0) and [[beta].sub.t] = [[beta].sub.0] [e.sup.tr] where [[beta].sub.0] and r are positive real numbers. Therefore [b.sub.t] = ([V.sub.t] - [a.sub.t] [j.sub.t](X)) [[beta].sup.-1.sub.t]. We interpret the above in the sense of expectation i.e given u [cross product] [psi](f) in the exponential domain of H [cross product] [GAMMA], where we will always assume u [not equal to] 0 so that [parallel]u [cross product] [psi](f)[parallel] [not equal to] 0,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

(i.e the value process is always in reference to a particular quantum mechanical state, so we can eventually reduce to real numbers) and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

As in the classical case we assume that the portfolio ([a.sub.t], [b.sub.t]), t [member of] [0, T] is self-financing i.e

d[V.sub.t] = [a.sub.t] [dj.sub.t](X) + [b.sub.t] d[[beta].sub.t]

or equivalently

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

Remark 2.1 The fact that the value process (and all other operator processes [X.sub.T] appearing in this paper) is always in reference to a particular quantum mechanical state, allows for a direct translation of all classical financial concepts described in Section 1 to the quantum case by considering the expectation (or matrix element) < u [cross product] [psi](f), [X.sub.t] u [cross product] [psi](f) > of the process at each time t. If the process is classical (i.e, if [X.sub.t] [member of] [??]) then we may divide out [[parallel]u [cross product] [psi](f)[parallel].sup.2] and everything is reduced to the classical case described in Section 1.

Lemma 2.1 Let [j.sub.t](X) = [U.sup.*.sub.t] X [cross product] 1 [U.sub.t] where {[U.sub.t] / t [greater than or equal to] 0} is the solution of (0.1). If

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.],

and

[theta] = i [H, X] - 1/2 {[L.sup.*] LX + X [L.sup.*] L - 2[L.sup.*] X L}

then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (0.2)

and for k [greater than or equal to] 2

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (0.3)

Proof. Equation (0.2) is a standard result of quantum flows theory (cf. [13]). To prove (0.3) we notice that for k = 2, using (0.2), the Ito table
``` . d[A.sup.[dagger].sub.t] d[[LAMBDA].sub.t]

d[A.sup.[dagger].sub.t] 0 0
d[[LAMBDA].sub.t] d[A.sup.[dagger].sub.t] d[[LAMBDA].sub.t]
d[A.sub.t] dt d[A.sub.t]
dt 0 0

. d[A.sub.t] dt

d[A.sup.[dagger].sub.t] 0 0
d[[LAMBDA].sub.t] 0 0
d[A.sub.t] 0 0
dt 0 0
```

and the homomorhism property [j.sub.t](x y) = [j.sub.t](x) [j.sub.t](y), we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

so (0.3) is true for k = 2. Assuming (0.3) to be true for k we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Thus (0.3) is true for k + 1 also.

3. Derivation of the Quantum Black-Scholes Equation

In the spirit of the previous section, let [V.sub.t] := F(t, [j.sub.t](X)) where F : [0, T] x B(H [cross product] [GAMMA]) [right arrow] B(H [cross product] [GAMMA]) is the extension to self-adjoint operators x = [j.sub.t](X) of the analytic function F(t, x) = [[summation].sup.+[infinity].sub.n,k=0] [a.sub.n,k]([t.sub.0], [x.sub.0]) [(t - [t.sub.0]).sup.n] [(x - [x.sub.0]).sup.k], where x and [a.sub.n,k]([t.sub.0], [x.sub.0]) are in C, and for [lambda], [mu] [member of] {0, 1, ...}

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

and so, if 1 denotes the identity operator then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

Notice that for ([t.sub.0], [x.sub.0]) = (0, 0) we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

Proposition 3.1 (Quantum Black-Scholes Equation)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

(this is the quantum analogue of the classical Black-Scholes equation) and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

Proof. By Lemma 2.1 and the Ito table for quantum stochastic differentials

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

where [alpha], [[alpha].sup.[dagger]], [lambda] are as in Lemma 2.1. Thus

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (0.4)

where [theta] is as in Lemma 2.1. We can obtain another expression for d[V.sub.t] with the use of the self-financing property. We have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

which can be written as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (0.5)

Equating the coefficients of dt and the quantum stochastic differentials in (0.4) and (0.5) we obtain the desired equations.

4. The case S = 1: Quantum Brownian motion

Proposition 4.1 Let F be as in the previous section. If S = 1 then the equations of Proposition 3.1 combine into

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

with initial condition u(0, [j.sub.T] (X)) = [([j.sub.T] (X) - K).sup.+] where u(t, x) = F(T - t, x), g(x) = [[y.sup.*], x] [x, y], h(x) = x r and x, y [member of] B(H [cross product] [GAMMA])

Proof. If S = 1 then, in the notation of Lemma 2.1, [alpha] = [[L.sup.*], X], [[alpha].sup.[dagger]] = [X, L], [lambda] = 0, and [theta] = i [H, X] - 1/2 {[L.sup.*] L X + X [L.sup.*] L - 2 [L.sup.*] X L} and the equations of Proposition 3.1 reduce to

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

which are condensed into

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

and

[a.sub.1,0](t, [j.sub.t](X)) = [a.sub.t].

Upon substituting the second of the last two equations into the first one and simplifying we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

which can be written as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

with terminal condition [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]. Letting x = [j.sub.t](X), y = [j.sub.t](L) be arbitrary elements in B(H [cross product] [GAMMA]) and letting g(x) = [[y.sup.*], x] [x, y], h(x) = x r, we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

Letting u(t, x) := F(T - t, x), [u.sub.1 0](t, x) = -[F.sub.1 0](T - t, x), [u.sub.0 2](t, x) = [F.sub.0 2](T - t, x) and [u.sub.0 1](t, x) = [F.sub.0 1](T - t, x) we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

with u(0, [j.sub.T](X)) = [([j.sub.T](X) - K).sup.+].

5. The case S [not equal to] 1: Quantum Poisson Process

In this section we examine the equations of Proposition 3.1 under the assumption S [not equal to] 1.

Proposition 5.1 Let F be as in Section 3. If [X, S] = S then the equations of Proposition 3.1 combine into

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

with initial condition [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Proof. Since X is self-adjoint and S is unitary, assuming that [X, S] = S is equivalent to assuming that [lambda] = [S.sup.*] X S - X = 1 and the equations of Proposition 3.1 take the form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

which are satisfied if

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

and [a.sub.t] = [[summation].sup.+[infinity].sub.k=1] [a.sub.0,k](t, [j.sub.t](X)) which, if substituted in the previous one, yields

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

But

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Letting x = [j.sub.t](X), y = [j.sub.t](L), z = [j.sub.t](H), h(x) = x r and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

using the notation of the previous section we obtain the Black-Scholes equation for the case S [not equal to] 1 as stated in the Proposition.

6. Solution of the Quantum Brownian Motion Black-Scholes Equation

To solve the Quantum Brownian motion Black-Scholes equation we assume that [j.sub.t]([X.sup.2]) = [j.sub.t]([[L.sup.*], X] [X, L]) which is the same as [X.sup.2] = [[L.sup.*], X] [X, L]. Since X = [X.sup.*], it follows that [[L.sup.*], X] = [[X, L].sup.*] and so letting [empty set](X) = [X, L] we find [X.sup.2] = [empty set][(X).sup.*] [empty set](X) i.e [empty set](X) = W X which implies that [X, L] = W X and [[L.sup.*], X] = X [W.sup.*], where W is an arbitrary unitary operator acting on the system space. In this case equation (0.2) takes the form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

Lemma 6.1 If H > 0 is a bounded self-adjoint operator on a Hilbert space H then there exists a bounded self-adjoint operator A on H such that H = [e.sup.A].

Proof. Let H = [[integral].sup.b.sub.a] [lambda]d[E.sub.[lambda]], where [a, b] [subset] (0, + [infinity]) and a [less than or equal to] [parallel]H[parallel] [less than or equal to] b. Letting [lambda] = [e.sup.[mu]] we obtain H = [[integral].sup.ln b.sub.ln a] [mu] dF([mu]) where F([mu]) = E([e.sup.[mu]]). Thus H = [e.sup.A] where A = [[integral].sup.ln b.sub.ln a] [mu] dF([mu]) with [parallel]A[parallel] less than or equal to] max (|ln a|, | ln b|). To show that the family {F([mu])/ ln a [les than or equal to] [mu] [less than or equal to] ln b} is a resolution of the identity we notice that for h [member of] H and [lambda], [mu] [member of] [ln a, ln b] we have:

(i) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

(ii) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

(iii) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

(iv) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

(v) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

and the proof is complete.

The equation in Proposition 4.1 now has the form

[u.sub.1 0](t, x) = 1/2 [u.sub.0 2](t, x) [x.sup.2] + [u.sub.0 1](t, x) x r - u(t, x) r

with initial condition u(0, [j.sub.T] (X)) = [([j.sub.T] (X) - K).sup.+] where we may assume that x is a bounded self-adjoint operator. Since

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

and x = [j.sub.t](X) > 0, and K are invertible, we may let x = K [e.sup.z] where z is a bounded self-adjoint operator commuting with K, and obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Similarly

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

and so

[[omega].sub.0 2](t, z) - [[omega].sub.0 1](t, z) = [u.sub.0 2](t, x) [x.sup.2].

Finally

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

and so

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]. (0.6)

with initial condition [omega](0, [z.sub.T]) = [([j.sub.T](X)-K).sup.+] where [z.sub.T] is defined by K [e.sup.z]T = [j.sub.T](X).

Theorem 6.1 In analogy with the classical case presented in section 1, the solution of (0.6) is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.],

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

Proof. We have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.],

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.],

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.],

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Thus

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

It follows that A = B = 0 thus proving (0.6). Moreover, in order to prove that the initial condition is satisfied, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

But

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

and so [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]. Thus, it suffices to show that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

We have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Suppose that K [e.sup.[z.sub.T]] [greater than or equal to] K. Then [z.sub.T] [greater than or equal to] 0 and by the spectral resolution theorem [z.sup.2n+1.sub.T] = [[integral].sup.+[infinity].sub.0] [[lambda].sup.2n+1] d[E.sub.[lambda]]. So

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Similarly, if K [e.sup.[z.sub.T]] < K then [z.sub.T] < 0 and if we let [z.sub.T] = -[w.sub.T] where [w.sub.T] = [[integral].sup.+[infinity].sub.0] [lambda]d[E.sub.[lambda]] > 0, then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

and so, as before, [PHI](g(0, K [e.sup.[z.sub.T]])) = 1/2 - 1/2 x 1 = 0.

Corollary 6.1 The reasonable price for a quantum option is [omega](T, [z.sub.0]) where [omega] is as in Theorem 6.1 and [z.sub.0] is defined by X = K [e.sup.[z.sub.0]]. The associated quantum portfolio ([a.sub.t], [b.sub.t]) is given by

[a.sub.t] = [[omega].sub.01](t - T, [z.sub.t])

[b.sub.t] = ([omega](T - t, [z.sub.t]) - [a.sub.t] [j.sub.t](X)) [e.sup.-tr] [[beta].sub.0].sup.-1]

where [z.sub.t] is defined by [j.sub.t](X) = K [e.sup.[z.sub.t]]. (As in the classical case described in Section 1, a reasonable price is defined as one which when invested at time t = 0 in a mixed portfolio, allows the investor through a self-financing strategy to end up at time t = T with an amount of

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

which is the same as the payoff, had the option been purchased. Here, u [cross product] [psi](f) is any vector in the exponential domain of H - [cross product] [GAMMA]).

Proof. By Theorem 6.1, the reasonable price for a quantum option is [V.sub.0] = F(0, [j.sub.0](X)) = F(0, X) = u(T, X) = [omega](T, [z.sub.0]). The formulas for [a.sub.t] and [b.sub.t] follow from the definition of the portfolio, given in Section 2.

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Luigi Accardi

Centro Vito Volterra, Universita di Roma Tor Vergata

via Columbia 2, 00133 Roma, Italy

E-mail: accardi@volterra.mat.uniroma2.it

Andreas Boukas

Department of Mathematics and Natural Sciences

The American College of Greece, Aghia Paraskevi, Athens 15342, Greece

E-mail: andreasboukas@acgmail.gr
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