Stock markets and second quantization

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Stock markets and second quantization

Fabio Bagarello

Campobasso – Maggio 2007

Plan of the talk Why quantum . . . the model

the stochastic limit . . . The FPL approach A generalized model Last considerations

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I. Plan of the talk

1. why qm in stock markets

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1. why qm in stock markets

2. introducing the strategy: a first toy model

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the stochastic limit . . . The FPL approach A generalized model Last considerations

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I. Plan of the talk

1. why qm in stock markets

2. introducing the strategy: a first toy model 3. the model...

the model

the stochastic limit . . . The FPL approach A generalized model Last considerations

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1. why qm in stock markets

2. introducing the strategy: a first toy model 3. the model...

4. ...its stochastic limit

Plan of the talk Why quantum . . . the model

the stochastic limit . . . The FPL approach A generalized model Last considerations

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I. Plan of the talk

1. why qm in stock markets

2. introducing the strategy: a first toy model 3. the model...

4. ...its stochastic limit

5. a fixed point-like approximation

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1. why qm in stock markets

2. introducing the strategy: a first toy model 3. the model...

4. ...its stochastic limit

5. a fixed point-like approximation 6. an interesting generalization...

Plan of the talk Why quantum . . . the model

the stochastic limit . . . The FPL approach A generalized model Last considerations

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I. Plan of the talk

1. why qm in stock markets

2. introducing the strategy: a first toy model 3. the model...

4. ...its stochastic limit

5. a fixed point-like approximation 6. an interesting generalization...

7. ...and conclusions

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kets

Our ideas of a closed market:

• the total number of shares is conserved;

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II. Why quantum mechanics in stock mar- kets

Our ideas of a closed market:

• the total number of shares is conserved;

• the total amount of cash is conserved;

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kets

Our ideas of a closed market:

• the total number of shares is conserved;

• the total amount of cash is conserved;

• the price of the share does not change continu- ously but with discrete steps, multiples of a given monetary unit;

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II. Why quantum mechanics in stock mar- kets

Our ideas of a closed market:

• the total number of shares is conserved;

• the total amount of cash is conserved;

• the price of the share does not change continu- ously but with discrete steps, multiples of a given monetary unit;

• the number of shares is a discrete quantity;

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kets

Our ideas of a closed market:

• the total number of shares is conserved;

• the total amount of cash is conserved;

• the price of the share does not change continu- ously but with discrete steps, multiples of a given monetary unit;

• the number of shares is a discrete quantity;

• the price of the share is fixed by the tendency of the market to sell a share, i.e. by the market sup- ply: the price increases when the market supply decreases and viceversa;

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Then operator theory is a natural framework. It can be used

(a.) to discuss the existence of conserved quantities and

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(a.) to discuss the existence of conserved quantities and

(b.) to find the differential equations of motion which drive the portfolio of each trader.

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Then operator theory is a natural framework. It can be used

(a.) to discuss the existence of conserved quantities and

(b.) to find the differential equations of motion which drive the portfolio of each trader.

For that we need a certain self-adjoint operator H, the hamiltonian.

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(a.) to discuss the existence of conserved quantities and

(b.) to find the differential equations of motion which drive the portfolio of each trader.

For that we need a certain self-adjoint operator H, the hamiltonian.

In particular:

• H determines the dynamics of the observables of the system:

X ∈ A ⇒ X(t) = e^{i Ht}Xe^{−i Ht}

and X(t) solves the Heisemberg equation of motion X(t) = i [H, X(t)]˙

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Then operator theory is a natural framework. It can be used

(a.) to discuss the existence of conserved quantities and

(b.) to find the differential equations of motion which drive the portfolio of each trader.

For that we need a certain self-adjoint operator H, the hamiltonian.

In particular:

• H determines the dynamics of the observables of the system:

X ∈ A ⇒ X(t) = e^{i Ht}Xe^{−i Ht}

and X(t) solves the Heisemberg equation of motion X(t) = i [H, X(t)]˙

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if a and a^† are such that [a, a^†] = 11 and ϕ0 ∈ H, aϕ₀ = 0, then we put

N = a^†a, ϕ_n = (a^†)ⁿ

√n! ϕ₀ ⇒ Nϕ_n = nϕ_n, N(aϕn) = (n − 1)(aϕn), N(a^†ϕn) = (n + 1)(a^†ϕn).

a annihilation operator, a^† creation operator;

N = a^†a number operator

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Our prototype model:

1. Our market consists of L traders exchanging a single kind of share;

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1. Our market consists of L traders exchanging a single kind of share;

2. a trader can only interact with a single other trader: i.e. the traders feel only a two-body in- teraction;

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Our prototype model:

1. Our market consists of L traders exchanging a single kind of share;

2. a trader can only interact with a single other trader: i.e. the traders feel only a two-body in- teraction;

3. the traders can only buy or sell one share in any single transaction;

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1. Our market consists of L traders exchanging a single kind of share;

2. a trader can only interact with a single other trader: i.e. the traders feel only a two-body in- teraction;

3. the traders can only buy or sell one share in any single transaction;

4. the price of the share changes with discrete steps, multiples of a given monetary unit (that we take

= 1 here);

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Our prototype model:

1. Our market consists of L traders exchanging a single kind of share;

2. a trader can only interact with a single other trader: i.e. the traders feel only a two-body in- teraction;

3. the traders can only buy or sell one share in any single transaction;

4. the price of the share changes with discrete steps, multiples of a given monetary unit (that we take

= 1 here);

5. as already mentioned, the price of the share is fixed by the market supply (expressed in term of natural numbers);

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The formal hamiltonian of the model is

8>

><

>>

:

H = H˜ 0+ ˜HI, where H₀ =PL

l =1αla^†_lal +PL

l =1βlc_l^†cl + o^†o + p^†p H˜_I =PL

i ,j =1p_{i j}



a^†_ia_j(cic_j^†)^Pˆ+ aia_j^†(cj c_i^†)^Pˆ



+ (o^†p + p^†o), (1)

where ˆP = p^†p and p_{i i} = 0 (a trader does not inter- act with himself!).

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The formal hamiltonian of the model is

8>

><

>>

:

H = H˜ 0+ ˜HI, where H₀ =PL

l =1αla^†_lal +PL

l =1βlc_l^†cl + o^†o + p^†p H˜_I =PL

i ,j =1p_{i j}



a^†_ia_j(cic_j^†)^Pˆ+ aia_j^†(cj c_i^†)^Pˆ



+ (o^†p + p^†o), (1)

where ˆP = p^†p and p_{i i} = 0 (a trader does not inter- act with himself!).

a^]_l → number of shares operator, p^] → price operator,

c_l^] → cash operator and o^] → supply operator.

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The formal hamiltonian of the model is

8>

><

>>

:

H = H˜ 0+ ˜HI, where H₀ =PL

l =1αla^†_lal +PL

l =1βlc_l^†cl + o^†o + p^†p H˜_I =PL

i ,j =1p_{i j}



a^†_ia_j(cic_j^†)^Pˆ+ aia_j^†(cj c_i^†)^Pˆ



+ (o^†p + p^†o), (1)

where ˆP = p^†p and p_{i i} = 0 (a trader does not inter- act with himself!).

a^]_l → number of shares operator, p^] → price operator,

c_l^] → cash operator and o^] → supply operator.

We assume the following CCR:

[al, a^†_n] = [cl, c_n^†] = δl nI, [p, p^†] = [o, o^†] = I, (2)

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The states of the market are

ω{n};{k};O;M( . ) =< ϕ{n};{k};O;M, . ϕ{n};{k};O;M >, (3)

where {n} = n₁, n2, . . . , nL, {k} = k₁, k2, . . . , kL and

ϕ{n};{k};O;M := (a^†₁)ⁿ¹· · · (a^†_L)^nL(c₁^†)^k1· · · (c_L^†)^kL(o^†)^O(p^†)^M

√n₁! . . . nL! k1! . . . kL! O! M! ϕ₀ (4)

are the eigenstates of c_l^†cl, a^†_lal, o^†o, p^†p. Here ϕ₀ is the vacuum of the model: a_jϕ0 = c_jϕ0 = pϕ0 = oϕ₀ = 0, for j = 1, 2, . . . , L.

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The states of the market are

ω{n};{k};O;M( . ) =< ϕ{n};{k};O;M, . ϕ{n};{k};O;M >, (3)

where {n} = n₁, n2, . . . , nL, {k} = k₁, k2, . . . , kL and

ϕ{n};{k};O;M := (a^†₁)ⁿ¹· · · (a^†_L)^nL(c₁^†)^k1· · · (c_L^†)^kL(o^†)^O(p^†)^M

√n₁! . . . nL! k1! . . . kL! O! M! ϕ₀ (4)

are the eigenstates of c_l^†cl, a^†_lal, o^†o, p^†p. Here ϕ₀ is the vacuum of the model: a_jϕ0 = c_jϕ0 = pϕ0 = oϕ₀ = 0, for j = 1, 2, . . . , L.

The role of the state:

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The states of the market are

ω{n};{k};O;M( . ) =< ϕ{n};{k};O;M, . ϕ{n};{k};O;M >, (3)

where {n} = n₁, n2, . . . , nL, {k} = k₁, k2, . . . , kL and

ϕ{n};{k};O;M := (a^†₁)ⁿ¹· · · (a^†_L)^nL(c₁^†)^k1· · · (c_L^†)^kL(o^†)^O(p^†)^M

√n₁! . . . nL! k1! . . . kL! O! M! ϕ₀ (4)

are the eigenstates of c_l^†cl, a^†_lal, o^†o, p^†p. Here ϕ₀ is the vacuum of the model: a_jϕ0 = c_jϕ0 = pϕ0 = oϕ₀ = 0, for j = 1, 2, . . . , L.

The role of the state:

(1) it "projects" from quantum to classical dynam- ics;

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The states of the market are

ω{n};{k};O;M( . ) =< ϕ{n};{k};O;M, . ϕ{n};{k};O;M >, (3)

where {n} = n₁, n2, . . . , nL, {k} = k₁, k2, . . . , kL and

ϕ{n};{k};O;M := (a^†₁)ⁿ¹· · · (a^†_L)^nL(c₁^†)^k1· · · (c_L^†)^kL(o^†)^O(p^†)^M

√n₁! . . . nL! k1! . . . kL! O! M! ϕ₀ (4)

are the eigenstates of c_l^†cl, a^†_lal, o^†o, p^†p. Here ϕ₀ is the vacuum of the model: a_jϕ0 = c_jϕ0 = pϕ0 = oϕ₀ = 0, for j = 1, 2, . . . , L.

The role of the state:

(1) it "projects" from quantum to classical dynam- ics;

(2) it fixes the initial conditions.

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The meaning of the hamiltonian:

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The meaning of the hamiltonian:

Contribution o^†p: when the supply increases then the price must decrease.

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The meaning of the hamiltonian:

Contribution o^†p: when the supply increases then the price must decrease.

Contribution

 a_i^†c_i^Pˆ

 

aj (c_j^†)^Pˆ



: trader τ_i has one more shares in his portfolio but, at the same time, ˆP less units of cash. Clearly, trader τ_j behaves in the opposite way: he loses one share because of a_j but his cash increases because of (c_j^†)^Pˆ.

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The meaning of the hamiltonian:

Contribution o^†p: when the supply increases then the price must decrease.

Contribution

 a_i^†c_i^Pˆ

 

aj (c_j^†)^Pˆ



: trader τ_i has one more shares in his portfolio but, at the same time, ˆP less units of cash. Clearly, trader τ_j behaves in the opposite way: he loses one share because of a_j but his cash increases because of (c_j^†)^Pˆ.

Since c_j^Pˆ has a clear economical meaning but is a tricky objectmathematically speaking, in [B.,J.Phys.A, 2006] we replaced c_j^Pˆ with c_j^M, M = ω{n};{k};O;M( ˆP ).

This produces an effective hamiltonian in which the price of the share is frozen to M.

However, an interesting dynamical still exists:

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The following operators are integrals of motion:

N =ˆ XL

i =1

a_i^†ai, K =ˆ XL

i =1

c_i^†ci and Γ = oˆ ^†o + p^†p,

since [H, ˆN] = [H, ˆΓ ] = [H, ˆK] = 0, as well as ˆQ_j = a^†_j a_j + _M¹ c_j^†c_j, for j = 1, 2, . . . , L.

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N =ˆ X

i =1

a_i^†ai, K =ˆ X

i =1

c_i^†ci and Γ = oˆ ^†o + p^†p,

since [H, ˆN] = [H, ˆΓ ] = [H, ˆK] = 0, as well as ˆQ_j = a^†_j a_j + _M¹ c_j^†c_j, for j = 1, 2, . . . , L.

The relevant differential equations are the following ( _{d Xl}

d t = i (βl − αl)Xl + 2i X^∞(2ˆnl − Ql)

d ˆnl

d t = 2i



XlX^∞† − X^∞X_l^†

 (5)

where X_l = a_lc_l^† (M = 1 here) and X^∞ = τ − limL→∞ p˜

L

PL

i =1Xi, τ being the strong topology (wrt F ). This is a closed set and can be analytically solved, see [Bag.,J.Phys.A,2006], [Bag. Wascom 2005].

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The following operators are integrals of motion:

N =ˆ XL

i =1

a_i^†ai, K =ˆ XL

i =1

c_i^†ci and Γ = oˆ ^†o + p^†p,

since [H, ˆN] = [H, ˆΓ ] = [H, ˆK] = 0, as well as ˆQ_j = a^†_j a_j + _M¹ c_j^†c_j, for j = 1, 2, . . . , L.

The relevant differential equations are the following ( _{d Xl}

d t = i (βl − αl)Xl + 2i X^∞(2ˆnl − Ql)

d ˆnl

d t = 2i



XlX^∞† − X^∞X_l^†

 (5)

where X_l = a_lc_l^† (M = 1 here) and X^∞ = τ − limL→∞ p˜

L

PL

i =1Xi, τ being the strong topology (wrt F ). This is a closed set and can be analytically solved, see [Bag.,J.Phys.A,2006], [Bag. Wascom 2005].

Here we are more interested in improving this model by considering the time evolution of the price of the

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We will focus our attention on a single trader, τ , which interact with an ensemble of other traders in a way that extends the interaction introduced before.

We call system, S, all the dynamical quantities which refer to a fixed trader τ :

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III. the model

We will focus our attention on a single trader, τ , which interact with an ensemble of other traders in a way that extends the interaction introduced before.

We call system, S, all the dynamical quantities which refer to a fixed trader τ :

shares number operators: a, a^† and ˆn = a^†a, cash operators: c, c^† and ˆk = c^†c

price operators of the shares: p, p^† and ˆP = p^†p.

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We will focus our attention on a single trader, τ , which interact with an ensemble of other traders in a way that extends the interaction introduced before.

We call system, S, all the dynamical quantities which refer to a fixed trader τ :

shares number operators: a, a^† and ˆn = a^†a, cash operators: c, c^† and ˆk = c^†c

price operators of the shares: p, p^† and ˆP = p^†p.

We call reservoir, R, all the other quantities:

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III. the model

We will focus our attention on a single trader, τ , which interact with an ensemble of other traders in a way that extends the interaction introduced before.

We call system, S, all the dynamical quantities which refer to a fixed trader τ :

shares number operators: a, a^† and ˆn = a^†a, cash operators: c, c^† and ˆk = c^†c

price operators of the shares: p, p^† and ˆP = p^†p.

We call reservoir, R, all the other quantities:

shares number operators: A_k, A^†_k and ˆNk = A^†_k Ak

cash operators: C_k, C_k^† and ˆK_k = C_k^†C_k (both re- lated to the other traders). Here k ∈ Λ and Λ = {1, 2, . . . , N − 1}.

supply of the market operators: ok, o_k^† and ˆOk = o_k^†o_k, k ∈ Λ.

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>>

>>

><

>>

>>

>:

H = H0 + λ HI, where H₀ = ωan + ωˆ ck + ωˆ pP +ˆ

+P

k∈Λ



ΩA(k) ˆN_k + ΩC(k) ˆK_k + ΩO(k) ˆO_k

 H_I =



z^†Z(f ) + z Z^†(f )



+ (p^†o(g) + p o^†(g))

(1)

Here ωa, ωc and ωp are positive real numbers and Ω_A(k), Ω_C(k) and Ω_O(k) are real valued non nega- tive functions. We have also introduced the following smeared fields of the reservoir:

8>

>>

><

>>

>>

:

Z(f ) = P

k∈ΛZ_k f (k) = P

k∈ΛA_k C_k^†

Pˆ

f (k), Z^†(f ) = P

k∈ΛZ_k^†f (k) = P

k∈ΛA^†_k C_k^Pˆf (k) o(g) = P

k∈Λo_k g(k) o^†(g) = P

k∈Λo_k^†g(k),

as well as the operators z = a c^†Pˆ, Z_k = Ak C_k^†

Pˆ

and their conjugates. They all have a clear economical meaning.

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Definition of c^†Pˆ

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Definition of c^†Pˆ

Let us, for simplicity, consider two operators c and p, with [c, c^†] = [p, p^†] = 11, and let ϕ0 be the vacuum:

c ϕ0 = pϕ0 = 0. Then ϕ_{k ,m} = 1

√k ! m! (c^†)^k(p^†)^mϕ₀,

where k, m ≥ 0, is an eigenstate of ˆk = c^†c and P = pˆ ^†p: ˆk ϕ_{k ,m} = kϕ_{k ,m} and ˆP ϕ_{k ,m} = mϕ_{k ,m}.

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Definition of c^†Pˆ

Let us, for simplicity, consider two operators c and p, with [c, c^†] = [p, p^†] = 11, and let ϕ0 be the vacuum:

c ϕ0 = pϕ0 = 0. Then ϕ_{k ,m} = 1

√k ! m! (c^†)^k(p^†)^mϕ₀,

where k, m ≥ 0, is an eigenstate of ˆk = c^†c and P = pˆ ^†p: ˆk ϕ_{k ,m} = kϕ_{k ,m} and ˆP ϕ_{k ,m} = mϕ_{k ,m}. Since

c^†ϕk ,m =p

k + 1ϕk +1,m, (c^†)²ϕk ,m = q

(k + 1)(k + 2)ϕk +2,m,

and so on, for all k and m ≥ 0, we put

c^†Pˆϕk ,m :=

( ϕ_{k ,m}, if m = 0,

p(k + 1)(k + 2) · · · (k + m) ϕ_{k +m,m}, if m > 0,

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Then c^Pˆ is just the adjoint of c^†Pˆ: c^Pˆ :=

 c^†Pˆ

†

.

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Then c^Pˆ is just the adjoint of c^†Pˆ: c^Pˆ :=

 c^†Pˆ

†

. We also deduce that

8>

>>

><

>>

>>

:

[ ˆP , c^Pˆ] = [ ˆP , c^†Pˆ] = 0, hk , cˆ ^Pˆ

i

= h

c^†c , c^Pˆ i

= − ˆP c^Pˆ = −c^PˆPˆ hk , cˆ ^†Pˆ

i

= ˆP c^†Pˆ = c^†PˆP ,ˆ

where, for instance, [ˆk , c^Pˆ] = − ˆP c^Pˆ is an extended version of [ˆk , c^l] = −l c^l.

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sumed:

[c, c^†] = [p, p^†] = [a, a^†] = 11, (2)

[oi, o_j^†] = [Ai, A_j^†] = [Ci, C_j^†] = δi ,j11 (3) which implies

[ ˆKk, C_q^Pˆ] = − ˆP C_q^Pˆδk ,q, [ ˆKk, C_q^†Pˆ] = ˆP C_q^†Pˆδk ,q

(4)

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back to the model

The following non trivial commutation rules are as- sumed:

[c, c^†] = [p, p^†] = [a, a^†] = 11, (2)

[oi, o_j^†] = [Ai, A_j^†] = [Ci, C_j^†] = δi ,j11 (3) which implies

[ ˆKk, C_q^Pˆ] = − ˆP C_q^Pˆδk ,q, [ ˆKk, C_q^†Pˆ] = ˆP C_q^†Pˆδk ,q

(4) The functions f (k) and g(k) in (1) are sufficiently regular to allows for the smeared fields to be well defined, as well as other relevant quantities we will obtain along the way.

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Quit

sumed:

[c, c^†] = [p, p^†] = [a, a^†] = 11, (2)

[oi, o_j^†] = [Ai, A_j^†] = [Ci, C_j^†] = δi ,j11 (3) which implies

[ ˆKk, C_q^Pˆ] = − ˆP C_q^Pˆδk ,q, [ ˆKk, C_q^†Pˆ] = ˆP C_q^†Pˆδk ,q

(4) The functions f (k) and g(k) in (1) are sufficiently regular to allows for the smeared fields to be well defined, as well as other relevant quantities we will obtain along the way.

Remark:– Of course, since τ can be chosen arbitrarily, the asymmetry of the model is just apparent. In fact, changing τ , we will be able, in principle, to find the

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time evolution of the portfolio of each trader of the stock market.

If we now define N := ˆˆ n+X

k∈Λ

Nˆk, K := ˆˆ k +X

k∈Λ

Kˆk, Γ := ˆˆ P +X

k∈Λ

Oˆk

then we find that ˆN, ˆK and ˆΓ are constants of motion:

[H, ˆN] = [H, ˆK] = [H, ˆΓ ] = 0.

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If we now define N := ˆˆ n+X

k∈Λ

Nˆk, K := ˆˆ k +X

k∈Λ

Kˆk, Γ := ˆˆ P +X

k∈Λ

Oˆk

then we find that ˆN, ˆK and ˆΓ are constants of motion:

[H, ˆN] = [H, ˆK] = [H, ˆΓ ] = 0.

Our main output is to recover the equations of motion for the portfolio of the trader τ , defined as

Π(t) = ˆˆ P (t) ˆn(t) + ˆk (t). (5)

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time evolution of the portfolio of each trader of the stock market.

If we now define N := ˆˆ n+X

k∈Λ

Nˆk, K := ˆˆ k +X

k∈Λ

Kˆk, Γ := ˆˆ P +X

k∈Λ

Oˆk

then we find that ˆN, ˆK and ˆΓ are constants of motion:

[H, ˆN] = [H, ˆK] = [H, ˆΓ ] = 0.

Our main output is to recover the equations of motion for the portfolio of the trader τ , defined as

Π(t) = ˆˆ P (t) ˆn(t) + ˆk (t). (5) It is not surprising that this cannot be done exactly so that some perturbative technique is needed. We will adopt here two different, and orthogonal, strategies:

the stochastic limit approach and a fixed point-like (FPL) scheme.

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The first ingredient of the SL is the free evolution of the interaction hamiltonian: H_I(t) := e^{i H0t}H_Ie^{−i Ht}, which turns out to be

H_I(t) = z^†Z(f e^{i t ˆεZ})+z Z^†(f e^{−i t ˆεZ})+p^†o(g e^{i tε0})+p o^†(g e^{−i tε0}),

where we have defined

ˆ

ε_Z(k) := ˆP (ΩC(k) − ωc) − (ΩA(k) − ωa), ε_O(k) := ωp−ΩO(k)

(which will be assumed to have zero only at most on a set of zero measure) and, for instance, Z(f e^{i t ˆεZ}) = P

k∈Λf (k) e^{i t ˆεZ(k)}Zk.