Solving quantum stochastic LQR optimal control problem in Fock space and its application in finance

Computers and Mathematics with Applications xxx (xxxx) xxx

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Solving quantum stochastic LQR optimal control problem in Fock space and its application in finance S. Yaghobipour a , M. Yarahmadi b , a b

∗

Department of Mathematics and Computer Science, Lorestan University, Iran Applied Mathematics, Department of Mathematics and Computer Science, Lorestan University, Iran

article

info

Article history: Received 28 January 2019 Received in revised form 23 September 2019 Accepted 15 December 2019 Available online xxxx Keywords: Operator-valued stochastic process HJB equation Portfolio selection quantum stochastic LQR

a b s t r a c t This paper is an attempt for solving operator-valued quantum stochastic optimal control problems, in Fock space. For this purpose, the dynamics of the classical system state is described by Hudson–Parthasarathy type Quantum Stochastic Differential Equation (QSDE) in Fock space and then by associating a quadratic performance criterion with the QSDE, a Quantum Stochastic Linear Quadratic Regulator (QS-LQR) optimal control problem is formulated. Also, an algorithm for solving the QS-LQR optimal control problem is designed. For solving the resulting optimal control problem, a new HJB equation is obtained. Thereby, the operator valued control process is obtained. Two theorems are proved to facilitate the algorithm. In this paper, for the first time, the optimal strategy for trading stock is designed via the presented method. For this purpose, Merton portfolio allocation problem is solved. The simulation results show that portfolio optimal performances, minimum risk and maximum return are achieved via presented method. © 2019 Elsevier Ltd. All rights reserved.

1. Introduction Usually, the dynamics of the economic or finance systems are modeled via a Stochastic Differential Equation (SDE) [1]. Therefore, for controlling such a problem, one can use a Stochastic Optimal Control (SOC) method. Merton portfolio allocation problem is such a problem [2]. Depending on type of all aspects (the dynamics, the systems environment and the cost function) an SOC problem is designed. For instance, an optimal control problem in which the system dynamics is described by a set of linear differential equations and a quadratic cost function is called the Linear Quadratic (LQ) optimal control problem. An applicable approach in LQ optimal control problem is Linear Quadratic Regulator (LQR) optimal control problem. In this design technique, the optimal control problem is transformed to a system of ordinary differential equations which is called Riccati equation [3]. Also, in this method the practiced feedback gain is provided. Also, if the state variable Xt is a stochastic variable, then a probability distribution via p(Xt |Y0:t ) is used instead of Xt , where Y0:t denotes all the previous observations of Xt based on Bayesian method. Also, usually, some of the stochastic differential equation parameters are unknown. In such cases, the learning techniques on finite/infinite horizon [4], the partial observable problem solving, [5–7] or the joint inference and control problem approaches [8] can be useful. The aim of an SOC problem solving is to obtain stochastic optimal state process of system controlled by stochastic optimal control process such that an expectational performance criterion is optimized [9]. The SOC problems can usually be solved by two conventional methods, using the Pontryagin Maximum Principle (PMP) which is based on the ∗ Corresponding author. E-mail addresses: [email protected] (S. Yaghobipour), [email protected] (M. Yarahmadi). https://doi.org/10.1016/j.camwa.2019.12.016 0898-1221/© 2019 Elsevier Ltd. All rights reserved.

Please cite this article as: S. Yaghobipour and M. Yarahmadi, Solving quantum stochastic LQR optimal control problem in Fock space and its application in finance, Computers and Mathematics with Applications (2019), https://doi.org/10.1016/j.camwa.2019.12.016.

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S. Yaghobipour and M. Yarahmadi / Computers and Mathematics with Applications xxx (xxxx) xxx

dynamic programming method such that the calculus of variations that yields a pair of ordinary differential equations [2] and considering the corresponding HJB equation [10–12]. For solving the HJB equation, discretizing method or direct numerical approaches are used, commonly. For example, the finite-difference and finite-element methods are used to obtain the transition probabilities of the Markov chain from the HJB equation in [13,14]. Additionally, path integral method can be a good choice. In this method, the non-linear HJB equation is transformed into a linear equation by a log transformation [5,15]. Indeed, expectation values can be computed under a forward diffusion process, instead of the usual backward integration of the HJB equation . Moreover, the stochastic integration can be described by a path integral [16]. The path integral leads us to non-linear Kalman filters method [17] and contributions in hidden Markov models [18]. This path integral can be approximated by Monte Carlo sampling [19] and variational approximation [20]. Apart from all classical methods, for solving the stochastic control problems, quantum mechanics as a flexible and powerful framework can be used to manipulate it as a quantum control system on a Hilbert space [21,22]. Moreover, the stochastic optimal control problems can be described in a linear formalism via the quantum mechanics, like as the path integral methods. Indeed, if the optimization methods for the classical control system are applied to the derivation of Hamilton–Jacobi equation then a type of Schrödinger equation is resulted [23]. In this method, it cannot be concluded that the resulting system has quantum behavior because, a white noise with complex values has been used. In addition to applying quantum formalism to HJB equation, the Brownian motion evolution (disturbance of the system) can be modeled by a type of the Schrödinger equation. Thereby, the corresponding optimal control problem can be formulated as a deterministic one that can be solved more easily [24]. Also, in [25], the system state is described by a Hudson–Parthasarathy type stochastic differential equation in Fock space and a control process is designed for minimizing the quadratic performance functional associated with a QSDE equation. But, a practical example has not been solved to guarantee the capability and efficiency of the presented method in [25]. In this paper, the quantum stochastic calculus is used for describing system dynamics, performance criterion and corresponding HJB equation such that the resulting method can be implemented, practically and the results of adopting quantum computation can be visible clearly, as well. For this purpose, the initial Hilbert space corresponding to the set of events is described and then the quantum stochastic LQR control problem is formulated for the resulting quantum system, in a symmetric Fock space over the initial Hilbert space. The resulting control problem is solved by considering a new corresponding Hamilton–Jacobi–Bellman (HJB) equation. Also, the optimal feedback control adapted process and Riccati equation is presented, consequently. As an financial application, Merton’s portfolio allocation problem [26] is solved via the presented quantum stochastic LQR method in Fock space and simulated for a case set of economic influence factors. The simulation results show the advantages and efficiency of the proposed method in determining the optimal operator-valued strategy for all of the events concerning the stock price, simultaneously. The paper is organized as follows; quantum stochastic calculus preliminary and the problem statement are presented in Sections 2 and 3, respectively. In Section 4, the presented quantum stochastic LQR optimal control problem in Fock Space is solved through two subsections, one presents the corresponding HJB equation and the other proposes a new quantum stochastic optimal control law in Fock space. Also, the Riccati equation appears in follows, consequently. Additionally, the algorithm of the proposed method is presented at the end of the section. Merton’s portfolio allocation problem is solved and simulated in Section 5 that is followed by conclusion in Section 6. 2. Quantum stochastic calculus preliminary Let H be a fixed complex separable Hilbert space and let ξ : FR+ → P (H) be a fixed R-valued observable with no jump points, i.e, ξ ({t }) = 0 for every t ≥ 0. By denoting the ranges of the projections ξ ([0, t ]), ξ ([s, t ]) and ξ ([t , ∞)) by Ht ] , H[s,t ] and H[t , respectively and considering the time partition 0 < t1 < t2 < · · · < tn < ∞, the following decomposition is defined [27]: H = Ht1 ] ⊕ H[t1 ,t2 ] ⊕ · · · ⊕ H[tj−1 ,tj ] ⊕ · · · ⊕ H[tn−1 ,tn ] ⊕ H[tn .

Let h0 be a fixed complex separable initial Hilbert space for describing events (projection operators) and observables (self-adjoint operators) concerning a system. If a noise process is modeled in the symmetric Fock space Γs (H) then the bounded and self-adjoint operators on the Hilbert space

˜ = h0 ⊗ Γs (H) H are used for describing the same objects concerning the noisy system [25]. Now, the following relations: for any 0 < s < t < ∞ is considered:

˜ 0 ] = h0 , H ˜ [t = Γs (H[t ), H

˜ t ] = h0 ⊗ Γs (Ht ] ), H ˜ [s,t ] = Γs (H[s,t ] ). H

Also, for any u ∈ H let ut ] = ξ ([0, t ])u, u[t = ξ ([t , ∞))u and u[s,t ] = ξ ([s, t ])u [25]. The elements of Γs (H) are defined for any u ∈ H as the following exponential (coherent) vectors e(u) =

∞ ⨁ 1 √ u⊗n , n! n=0

Please cite this article as: S. Yaghobipour and M. Yarahmadi, Solving quantum stochastic LQR optimal control problem in Fock space and its application in finance, Computers and Mathematics with Applications (2019), https://doi.org/10.1016/j.camwa.2019.12.016.

S. Yaghobipour and M. Yarahmadi / Computers and Mathematics with Applications xxx (xxxx) xxx

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where n is the number of particles which occupy a (bosonic) state by putting u⊗0 = 1 and 0! = 1. Indeed, the set of all the exponential vectors {e(u)|u ∈ H} is linearly independent and total set in Γs (H) [27], i.e :

Γs (H) = ⟨ {e(u)|u ∈ H} ⟩. Definition 1 (See [27]). A map m : t → mt from R+ into H is called ξ -martingale if mt ∈ Ht ] for every t and ξ ([0, s])mt = ms for all s < t where Ht ] = {ξ ((−∞, t ])u|u ∈ H}. Definition 2 (See [27]). Suppose m, m′ are two ξ -martingales then for any a > 0 we have ⟨mt , m′t ⟩ = ⟨ma , ξ ([0, t ])m′a ⟩ for all 0 ≤ t ≤ a. Therefore, there exists a complex valued measure ⟨⟨m, m′ ⟩⟩ in R+ that is defined as follows:

⟨⟨m, m′ ⟩⟩([0, t ]) = ⟨mt , m′t ⟩

(1)

for all t ⩾ 0. For example, if u ∈ H then ut ] = ξ ([0, t ])u, t ≥ 0 is a ξ -martingale. Also for three ξ -martingales u : t → ut ] , v : t → vt ] and m : t → mt ] , the following measures are defined.

⟨⟨u, v⟩⟩[0, t ] = ⟨ut ] , vt ] ⟩, ⟨⟨u, m⟩⟩[0, t ] = ⟨ut ] , mt ] ⟩, ⟨⟨m, v⟩⟩[0, t ] = ⟨mt ] , vt ] ⟩, ˜ = Γs (H). Let m be any ξ -martingale in H. Define the linear operators Example 1 (See [27]). Choose h0 = C so that H † Am (t) and Am (t) as follows: †

(i) dom(Am (t)) = dom(Am (t)) = ⟨{e(u)|u ∈ H}⟩, †

(ii) Am (t)e(u) = a† (mt )e(u), (iii) Am (t) = a(mt )e(u) = {⟨⟨m, u⟩⟩([0, t ])e(ut ] )}e(u[t ), u ∈ H, where a† (.) and a(.) are defined as q(u) = −p(iu), a(u) =

1 2

(q(u) + ip(u)), a† (u) = †

1 2

(q(u) − ip(u))

†

for observable p(u) ∈ Γs (H). The processes Am = {Am (t) |t ≥ 0} and Am = {Am (t) |t ≥ 0} are regular adapted processes † with respect to the triple (ξ , C, H). Am and Am are respectively called the creation and annihilation processes in Γs (H) associated with the ξ -martingale m [27]. Example 2 (See [27]). Denote the set of all bounded operators in H by B(H). Let H ∈ B(H) and Ht = H ξ ([0, t ]) = ξ ([0, t ])H, for all t ∈ R. The operator ΛH (t) in Γs (H) is defined as follows: (i) dom(ΛH (t)) = ⟨{e(u)|u ∈ H}⟩, (ii) ΛH (t)e(u) = λ(Ht )e(u) = λ(Ht )e(ut ] )e(u[t ), Ht + Ht∗

Ht − Ht∗

), λ† (Ht ) = λ(Ht∗ ). Since, the map ξ is continuous, thus ΛH = {ΛH (t)|t ≥ 0} is 2 2 a regular adapted processes with respect to (ξ , h0 , H). The process ΛH is called conservation process in Γs (H) associated with the Hilbert space H.

where λ(Ht ) = λ(

) + iλ(

The three processes in Examples 1 and 2 are called fundamental processes. Now, let D0 ⊆ h0 and M ⊂ H be linear manifolds such that ξ ([s, t ])u ∈ M whenever u ∈ M for all 0 ≤ s < t < ∞. Also, suppose (L, L† ) be an adjoint pair of (ξ , D0 , M)-adapted stochastically integrable processes [27], M be a fundamental process and t

∫

LdM ,

X (t) =

X † (t) =

t

∫

0

L† dM † (s).

(2)

0

˜ [27]. Then (X , X † ) is an adjoint pair of operators in H Corollary 1 (First Fundamental Lemma). According to Definition 2 and integrals in Eq. (2), for every (ξ , D0 , M)-adapted ˜ and for all f , g ∈ D0 and u, v ∈ H the following relation holds: stochastically integrable process Lt in H

⟨fe(u),

t

∫

L(s)dM(s)ge(v )⟩ = 0

t

∫

⟨fe(u), L(s)ge(v )⟩dµ(s),

(3)

0

†

where µ is equal to ⟨⟨u, m⟩⟩, ⟨⟨u, H v⟩⟩ or ⟨⟨m, v⟩⟩ for M = Am , ΓH or Am , respectively. Please cite this article as: S. Yaghobipour and M. Yarahmadi, Solving quantum stochastic LQR optimal control problem in Fock space and its application in finance, Computers and Mathematics with Applications (2019), https://doi.org/10.1016/j.camwa.2019.12.016.

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S. Yaghobipour and M. Yarahmadi / Computers and Mathematics with Applications xxx (xxxx) xxx Table 1 Determining dM1 dM2 .

˜ whose domain includes Let, Tξ = {⟨⟨m, m′ ⟩⟩ : m, m′ are two arbitrary ξ -martingales } and X0 be any operator in H D0 ⊗ {e(u)|u ∈ H} where D0 ⊂ h0 and for all f ∈ D0 and u ∈ M X0 fe(u) = (C0 f )e(u), where C0 is an operator in h0 whose domain is D0 . If Mj , 1 ≤ j ≤ n are fundamental processes, τj+n , for j = 1, 2, . . . , k are measures from set Tξ and Li , 1 ≤ i ≤ n + k are integrable adapted processes [27]. Then every regular (ξ , D0 , M)-adapted process X (t) is expressed as follows: X (t) = X0 +

n ∫ ∑ j=1

t

Lj dMj + 0

n+k ∫ ∑ j=n+1

t

Lj dτj , t ≥ 0.

(4)

0

Eq. (4) can be rewritten as dX =

n ∑

Lj dMj +

j=1

n+k ∑

Lj dτj , t ≥ 0.

(5)

j=n+1

Also, if for 1 ≤ j ≤ n + k, (Lj , L† ) is an adjoint pair of integrable adapted processes [27], then for each j dX † =

n ∑

†

Lj dMj +

j=1

n+k ∑

†

Lj dτ¯j ,

j=n+1

where τ¯j denotes complex conjugation of τj . Let M1 , M2 be fundamental processes in Γs (H) and B1 , B2 ∈ B(H). Then M1 M2 is a (ξ , H)-adapted process satisfying the following relation [28]: dM1 M2 = M1 dM2 + M2 dM1 + dM1 dM2 where dM1 dM2 is given by the defined multiplication in Table 1 . The following relation is more specialized version of quantum Ito’s formula which can be used for computational purposes: dXY = XdY + dXY + dXdY ,

(6)

˜. where X and Y are two regular (ξ , D0 , M)-adapted processes in H According to the assumption mentioned above, a quantum stochastic differential equation is suggested for describing the dynamics of a regular (ξ , D0 , M)-adapted process X = {X (t)|t ⩾ 0}. Let, L, S , H ∈ B(h0 ) where S is unitary and H is self-adjoint. Then, there exists a unique unitary (ξ , D0 , M)-adapted process U = {U(t)|t ⩾ 0} satisfying the following equation:

{

†

dU = {LdAm + (S − 1)dΛp − L∗ SdAm − (iH + 12 L∗ L)d⟨⟨m, m⟩⟩}U , U(0) = 1H˜ ,

(7)

˜ . Eq. (7) is interpreted as a Schrödinger equation in the presence of noise [27]. For where 1H˜ is identity operation in H presenting the Heisenberg picture of Eq. (7), put jt (X ) = U(t)∗ X ⊗ 1Γ U(t),

t⩾0

(8)

where U(t) is the solution of Eq. (7) and 1 is identity operator in Γs (H). Therefore, for any X ∈ B(h0 ), the quantum ˜ stochastic flow (or the quantum version of a Markov chain) {jt (X )|t ⩾ 0} is a regular (ξ , h0 , H)-adapted process in H satisfying the following stochastic differential equation †

djt (X ) = jt (S ∗ [X , L])dAm + jt (S ∗ XS − X )dΛp + jt ([L∗ , X ]S)dAm +jt (i[H , X ] − 21 {L∗ LX − XL∗ L − 2L∗ XL})d⟨⟨m, m⟩⟩,

(9)

where j0 (X ) = X ⊗ 1Γ . One can obtain Eq. (9) according to Eq. (7) and differentiating of Eq. (8), easily. Please cite this article as: S. Yaghobipour and M. Yarahmadi, Solving quantum stochastic LQR optimal control problem in Fock space and its application in finance, Computers and Mathematics with Applications (2019), https://doi.org/10.1016/j.camwa.2019.12.016.

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3. Problem statement Suppose that the evolution of the state of a controllable physical system, in the presence of noise, is described by the ˜ = h0 ⊗ Γs (H): following QSDE valued in H dX (t) = [A(t)X (t) + B(t)jt (f )u(t)]dt + [C (t)X (t) + D(t)u(t)]djt (f ),

(10)

where A(t), B(t), C (t) and D(t) are definite matrices with continuous functions components. The control process u = {u(t)|t ≥ 0} is an operator-valued (ξ , h0 , H)-adapted process in admissible control set A ⊂ H˜ . Here, j = {jt (f )|t ≥ 0, f ∈ B(H)} is a Quantum Stochastic Flow (QSF) satisfying the QSDE (9). Eq. (10) is a linear QSDE with control-dependent diffusion coefficients. Let ⟨⟨m, m⟩⟩ be a standard Lebesgue measure for a ξ -martingale m, according to QSDE. (9) and Eq. (10) we have: dX (t) = K (X , u, t)dt + M(X , u, t)dA†m + N(X , u, t)dAm + Z (X , u, t)dΛp

(11a)

where X (0) = X0 and K (X , u, t) = [A(t)X (t) + B(t)jt (f )u(t)] + [C (t)X (t)

+D(t)u(t)]jt (i[H , f ] − 12 {L∗ Lf − fL∗ L − 2L∗ fL}), M(X , u, t) = [C (t)X (t) + D(t)u(t)]jt (S ∗ [f , L]), N(X , u, t) = [C (t)X (t) + D(t)u(t)]jt ([L∗ , f ]S), Z (X , u, t) = [C (t)X (t) + D(t)u(t)]jt (S ∗ fS − f ).

(11b)

Additionally, Eq. (11a) is a linear QSDE with control-dependent diffusion coefficients [29]. According to classical stochastic control theory for linear regulator problem, a LQR optimal control problem refers to a linear system and quadratic criterion [1]. Therefore, in this paper, the following quadratic performance criterion is associated with the QSDE (10):

∫ T [ ] J(u) = E ⟨X (T )h, WT X (T )h⟩ + [⟨X (s)h, Qs X (s)h⟩ + ⟨u(s)h, Rs u(s)h⟩]dτm (s) ,

(12)

t

where h ∈ span{f ⊗ e(u)|f ∈ h0 , u ∈ H}, τ ∈ Tξ is a positive measure on [0, T ]. Also, the process W (t), invertible symmetric operator R(t) and symmetric operator Q (t) belong to B(h0 ⊗ Γs (H)) are (ξ , h0 , H)-adapted processes, for each t ∈ [0, T ]. Let R−1 = {R−1 (t) ∈ B(h0 ⊗ Γs (H))|0 ≤ t ≤ T } be an (ξ , h0 , H)-adapted process. The solution (ξ , h0 , H)-adapted process X = {X (t)|0 ≤ t ≤ T } of the QSDE (11) is called the response of the admissible control process u = {u(t)|t ≥ 0} and (X,u) is called an admissible pair. Maximization of J(u), prescribed in Eq. (12) is the main objective of the optimal control problem. Therefore, the considered quantum stochastic optimal control problem in the symmetric Fock space is formed as follows:

[

maxu∈A J(u) = E ⟨XT h, WT XT h⟩ +

∫T t

]

G(Xs , us , h)dτm (s) ,

subject to : † dXt = Kdt + MdAm + NdAm + ZdΛp , X (0) = X0 ,

(13)

where G(Xt , ut ) = ⟨X (t)h, QX (t)h⟩ + ⟨u(t)h, Ru(t)h⟩ and K , M , N , Z are as in Eq. (11b). 4. Solving quantum stochastic LQR optimal control problem in fock space In this section, a new solution method for the quantum stochastic LQR optimal control problem (13) in finite horizon is presented, which is facilitated by two theorems. In the first theorem, according to the quantum stochastic calculus mentioned in Section 2, the corresponding HJB equation is obtained. In the second theorem, the quantum stochastic optimal control process, based on the maximization of the quadratic performance criterion (12), is obtained and the Riccati equation associated with the quantum stochastic LQR optimal control problem (13) is yielded. Finally, a new algorithm for solving the quantum stochastic LQ optimal control problem in Fock space based on two presented theorems, is designed. 4.1. Corresponding HJB equation Consider the quantum stochastic LQR optimal control problem (13) and the following value function:

[

V (t , Xt ) = max E ⟨XT h, WT XT h⟩ + u∈A

T

∫

]

G(Xs , us , h)dτm (s) .

(14)

t

Please cite this article as: S. Yaghobipour and M. Yarahmadi, Solving quantum stochastic LQR optimal control problem in Fock space and its application in finance, Computers and Mathematics with Applications (2019), https://doi.org/10.1016/j.camwa.2019.12.016.

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S. Yaghobipour and M. Yarahmadi / Computers and Mathematics with Applications xxx (xxxx) xxx

Theorem 1.

Consider the quantum stochastic LQR optimal control problem in finite horizon (13), a HJB equation whose

solution optimizes the quadratic performance criterion (12) is

∂ V (t , Xt ) ∂ V (t , Xt ) + Kt + G(Xt , ut , h) = 0, ∂t ∂X

(15)

where V (t , Xt ) is the value function as (14). Proof. Let, 0 ≤ k ≤ T − t, the relation (14) can be rewritten as

[

V (t , Xt ) = max E ⟨XT h, WT XT h⟩ + u∈A

T

∫

G(Xs , us , h)dτm (s) + t +k t +k

G(Xs , us , h)dτm (s)

]

t

∫

[

t +k

∫

]

= E V (t + k, Xt +k ) + max G(Xs , us , h)dτm (s) u∈A t ∫ [ ] t +k ≥ E V (t + k, Xt +k ) + G(Xs , us , h)dτm (s) .

(16)

t

Let, M be a fundamental process. Since h ∈ span{f ⊗ e(u)|f ∈ h0 , u ∈ H} and τm ∈ Tξ then according to the relation (3), for an arbitrary constant control process {ut } = u ∈ A, the value function (14) can be rewritten as:

[

∫

T

∫

T

⟨uh, Rs h⟩dτm (s) ⟨Xs h, QXs h⟩dτm (s) + t t ∫ ∫ ] [ T T Rs dMuh⟩ . QdMXs h⟩ + ⟨uh, = E ⟨XT h, WT XT h⟩ + ⟨Xs h,

V (t , Xt ) = E ⟨XT h, WT XT h⟩ +

]

(17)

t

t

Since, Qt and Rt are stochastically integrable processes and M is a fundamental process, therefore T

∫

T

∫ Qs dM(s)

q(t) =

and

Rs dM(s)

r(t) = t

t

˜ . Thus, the value function (17) can be rewritten as: are two operators in H

[

V (t , Xt ) = E ⟨XT h, WT XT h⟩ +

∫

T

∫

T

⟨Xs h, QXs h⟩dτm (s) + ⟨uh, Rs h⟩dτm (s) t t ∫ ∫ ] [ T T Rs dMuh⟩ Qd MXs h⟩ + ⟨uh, = E ⟨XT h, WT XT h⟩ + ⟨Xs h, t [ ] t = E ⟨XT h, WT XT h⟩ + ⟨Xs h, qs Xs h⟩ + ⟨uh, rs uh⟩ .

]

(18)

Now, by applying the quantum Ito’s formula (6) and linearity of E(.), one can infer: dV = E

[

⟨dXh, qXh⟩ + ⟨Xh, [dqX + qdX + dqdX ]h⟩

+⟨dXh, [dqX + qdX + dqdX ]h⟩ + ⟨duh, ruh⟩

(19)

+⟨uh, [dru + rdu + dudr ]h⟩ + ⟨duh, [dru + rdu + drdu]h⟩

]

.

Since, control process u is constant thus du = 0. Also, processes X , q and r are positive process. So, all of the presented inner products in (19) are positive. Therefore, we have

[

]

dV ≥ E ⟨dXh, qXh⟩ + ⟨Xh, qdXh⟩ + ⟨Xh, dqXh⟩ + ⟨uh, druh⟩ .

(20)

Also, d⟨XT h, WT XT h⟩ dX

= 0,

d⟨uh, ruh⟩ dX

= 0,

d⟨XT h, WT XT h⟩ ds

= 0.

Please cite this article as: S. Yaghobipour and M. Yarahmadi, Solving quantum stochastic LQR optimal control problem in Fock space and its application in finance, Computers and Mathematics with Applications (2019), https://doi.org/10.1016/j.camwa.2019.12.016.

S. Yaghobipour and M. Yarahmadi / Computers and Mathematics with Applications xxx (xxxx) xxx

7

Therefore, inequality (20) can be written as:

) dX ( ) ds ] ⟨dXh, qXh⟩ + ⟨Xh, qdXh⟩ + ⟨Xh, dqXh⟩ + ⟨uh, druh⟩ dX ds ) ds ] [( ) dX ( + ⟨Xh, dqXh⟩ + ⟨uh, druh⟩ + 0 = E ⟨dXh, qXh⟩ + ⟨Xh, qdXh⟩ + 0 dX ds [ ( d⟨X h, W X h⟩ d⟨Xh, qXh⟩ d⟨uh, ruh⟩ ) T T T =E + + dX dX dX dX ( d⟨X h, W X h⟩ d⟨Xh, qXh⟩ d⟨uh, ruh⟩ ) ] T T T + + ds + ds ds ds ] [ ∂ V (t , X ) ∂ V (t , X ) dX + ds . =E ∂X ∂s for s ∈ [t , t + k]. Now, according to the linearity of E(.), we have: ∫ t +k ∫ t +k [∫ t +k ∂ V (t , X ) ∂ V (t , X ) ] dV ≥ E dX + ds . ∂X ∂s t t t dV ≥ E

[(

(21)

Therefore,

[∫

V (t + k, Xt +k ) ≥ V (t , Xt ) + E

t +k

t

∂ V (t , X ) dX + ∂X

∂ V (t , X ) ] ds . ∂s

t +k

∫ t

(22)

According to the inequalities (22) and (16), one can infer:

[

V (t , Xt ) ≥ E V (t + k, Xt +k ) +

[

≥ E V (t , Xt ) + E

[∫

∫

t t +k

t

t +k

G(Xs , u, h)dτm (s)

∂V dX + ∂X

t +k

∫ t

]

∂V ] ds + ∂s

t +k

∫

]

G(Xs , u, h)dτm (s) .

(23)

t

For the ξ -martingale m, measure ⟨⟨m, m⟩⟩ is the standard Lebesgue measure in R+ (i.e., d⟨⟨m, m⟩⟩ ≡ dt). Now, by replacing value of dX from Eq. (13) in Eq. (23) and E(V (t , Xt )) = V (t , Xt ) we have: 0≥E

t +k

[∫ t

∂V (Kds + MdA†m + NdAm + ZdΛp ) + ∂X

t +k

∫ t

∂V ds + ∂s

t +k

∫

]

Gs ds . t

Because the expectation of quantum stochastic fundamental processes (quantum noises) is equal to zero. Now, according to the mean-value theorem, one can infer:

∫ ∫ ∫ ] ] 1 [ t +k ∂ V ] 1 [ t +k ∂ V 1 [ t +k G(Xs , u, h)ds 0 ≥ lim E ds + lim E Ks ds + lim E k→0 k k→0 k k→0 k ∂s ∂X t t t ∂V ∂V ≥ + Kt + G(Xt , u, h). ∂t ∂X

(24)

Inequality (24) is satisfied for any u ∈ A, therefore HJB equation which maximizes the quadratic performance criterion (12) is

∂ V (t , Xt ) ∂ V (t , Xt ) + Kt + G(Xt , ut , h) = 0. □ ∂t ∂X 4.2. Quantum stochastic optimal control design in fock space In this section, the quantum stochastic optimal control process and the associated Riccati equation are obtained by considering the corresponding HJB equation (15) based on the following theorem. Theorem 2. Consider the stochastic differential equation (11), the associated performance criterion (12) is maximized by the following control process

(

)

u∗t = −R−1 Bjt (f ) + DV PX ,

(25)

where V = jt (i[H , f ] − 21 {L∗ Lf − fL∗ L − 2L∗ fL}) and P is the solution of the following Riccati differential equation:

˙ + O† P(t) + P(t)O + Q + (TP)† (R−1 )† TP = 0 P(t) for, O = [A(t) + B(t)jt (f )(−R

−1

−1

TP)] + [C (t) + D(t)(−R

(26)

TP)]V and T = Bjt (f ) + DV .

Please cite this article as: S. Yaghobipour and M. Yarahmadi, Solving quantum stochastic LQR optimal control problem in Fock space and its application in finance, Computers and Mathematics with Applications (2019), https://doi.org/10.1016/j.camwa.2019.12.016.

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S. Yaghobipour and M. Yarahmadi / Computers and Mathematics with Applications xxx (xxxx) xxx

Proof. Consider the HJB equation (15) corresponding to the quantum stochastic LQR optimal control problem in finite horizon (13). Let,

H(Vt , ut ) =

∂ V (t , Xt ) ∂ V (t , Xt ) + Kt + G(Xt , ut , h). ∂t ∂X

(27)

Let, the following quadratic function be considered as a value function: W (t , X ) = ⟨Xh, P(t)Xh⟩, where both P(t) and Xt are the operator-valued regular (ξ , h0 , H)-adapted processes. The value function W is satisfied in Eq. (27). Therefore : †

˙ H(Wt , ut ) = h† X † P(t)Xh + h† Kt P(t)Xh + h† X † P(t)Kt h + h† X † QXh + h† u† Ruh.

(28)

After replacing the value of Kt from Eq. (11) in to relation (28), we have:

( ) ˙ ∂ h† X † P(t)Xh

( ) ∂ h† (D + KV )† P(t)Xh

∂H = + ∂u ( ∂u ∂u ) ) ( ( ) † † ∂ h X P(t)(D + KV ) ∂ h† X † QXh ∂ h† u† Ruh + + + , ∂u ∂u ∂u

(29)

where D = [A(t)X (t) + B(t)jt (f )u(t)], K = [C (t)X (t) + D(t)u(t)] and V = jt (i[H , f ] − 12 {L∗ Lf − fL∗ L − 2L∗ fL}). Therefore,

( )† ∂H = h† Bjt (f ) + DV P(t)Xh ∂u ( ) +h† X † P(t) Bjt (f ) + DV + h† Ruh + h† u† Rh.

(30)

(

)

On the other hand, according to the theorem assumption, u∗t = −R−1 Bjt (f ) + DV PX . By replacing u∗ in relation (30) we have:

∂ H(Wt , u∗t ) = 0. ∂u Therefore u∗ is a quantum stochastic optimal control in Fock space Γ (H). Now, let T = Bjt (f ) + DV . According to Eq. (15), we have:

H(Wt , u∗t ) = S + h† u∗† Ru∗ h

= S + h† (−R−1 TPX )† R(−R−1 TPX )h = S + (Xh)† (TP)† (R−1 )† TP(Xh) = 0.

(31)

where †

˙ S = (Xh)† P(t)Xh + h† Kt P(t)(Xh) + (Xh)† P(t)Kt h + (Xh)† Q (Xh).

(32) ∗

According to Kt = [A(t)X (t) + B(t)jt (f )u(t)] + [C (t)X (t) + D(t)u(t)]V , by replacing the value of u (t), we have: Kt = [A(t)X (t) + B(t)jt (f )(−R−1 TPX )] + [C (t)X (t) + D(t)(−R−1 TPX )]V = ([A(t) + B(t)jt (f )(−R−1 TP)] + [C (t) + D(t)(−R−1 TP)]V )X = OX ,

(33)

where O = [A(t) + B(t)jt (f )(−R−1 TP)] + [C (t) + D(t)(−R−1 TP)]V . Therefore, by replacing Kt from Eq. (33), S in Eq. (32) becomes: †

˙ + h† Kt P(t)(Xh) + (Xh)† P(t)Kt h + (Xh)† Q (Xh) S = (Xh)† P(t)Xh ˙ = (Xh)† P(t)Xh + h† (OX )† P(t)(Xh) + (Xh)† P(t)(OX )h + (Xh)† Q (Xh) ( ) ˙ + O† P(t)O + Q (Xh). = (Xh)† P(t)

(34)

Now, according to S from Eq. (34), Eq. (31) is rewritten as follows:

H(Wt , u∗ ) = S + (Xh)† (TP)† (R−1 )† TP(Xh)

( ) ˙ + O† P(t)O + Q (Xh) + (Xh)† (TP)† (R−1 )† TP(Xh) = (Xh)† P(t) ( ) ˙ + O† P(t) + P(t)O + Q + (TP)† (R−1 )† TP (Xh) = 0, = (Xh)† P(t)

(35)

Please cite this article as: S. Yaghobipour and M. Yarahmadi, Solving quantum stochastic LQR optimal control problem in Fock space and its application in finance, Computers and Mathematics with Applications (2019), https://doi.org/10.1016/j.camwa.2019.12.016.

S. Yaghobipour and M. Yarahmadi / Computers and Mathematics with Applications xxx (xxxx) xxx

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since, Xh ̸ = 0, therefore a sufficient condition for holding the above equality, is that P(t) satisfy the following Riccati equation

˙ + O† P(t) + P(t)O + Q + (TP)† (R−1 )† TP = 0 P(t) and this completes the proof.

□

4.3. Algorithm of solving the quantum stochastic LQR optimal control problem in fock space Initial Quantum Space Prescribing a. Construct the initial Hilbert space corresponded to the set of events concerning the system. b. Design the quantum stochastic LQR problem in the form of (13) by describing quantum analogue of the given stochastic optimal control problem associated to the system, in symmetric Fock space over the initial Hilbert space described through part. a. Input: Specify all of the parameters of quantum stochastic LQR optimal control problem (13), X0 , P(0) = Q (0) = In and a final time T . Output: Admissible and Optimal pair (Xt∗ , u∗t ). Step 1. Solve the Riccati equation presented in (26) and find the symmetric square matrix P. Step 2. Compute the values of the quantum process {ut |t ∈ [0, T ]} by the control rule (25). Step 3. Compute the response of the control process, i.e, compute the solution (ξ , h0 , H)-adapted process X = {X (t)|0 ≤ t ≤ T } of Eqs. (11a) and (11b) Remark 1. By considering the discrete-time dynamics of the quantum system: dX (tk ) = [A(tk )X (tk ) + B(tk )jt (f )u(tk )]dt + [C (tk )X (tk ) + D(tk )u(tk )]djtk (f )

(36)

for k = 1, . . . , N, P(tk+1 ) is found iteratively forwards by the following dynamic Riccati equation [30]:

(

P(tk+1 ) = P(tk ) + O† P(tk ) + P(tk )O + Q (tk ) + (TP(tk ))† (R−1 )† TP(tk )

)

(37)

from terminal condition P(t0 ) = Q (0) = In . 5. Solving Merton portfolio allocation problem by quantum stochastic LQR control in fock space 5.1. Merton portfolio allocation problem According to the Merton definition, an investment in a mixed portfolio (i.e a part is invested in stock and reminding part in bond) at time t = 0, should lead an trader to end up at time t = T with a profit by an optimal strategy. It is assumed that the value of the portfolio is reduced with respect to the consumption rate. The goal is that satisfaction value of consuming be maximized. On the other hand, the minimization of the capital consumption (cost of the investment) is the main aim of this problem. Therefore, by solving the problem, the invested part in stock (risky asset), the invested part in pond (riskless asset) and capital consumption value can be computed. Suppose (u1 (t), α (t)) for t ∈ [0, T ] is a pair of trading strategies where α (t) = 1 − u1 (t) (i.e a portion u1 (t) is invested in the stock at time t and a portion α (t) is invested in bond at the same time). Additionally, let u2 (t) for t ∈ [0, T ] is capital consumption value at time t. Therefore, the value of the portfolio at time t is given by

wt = u1 (t)Xt + [1 − u1 (t)]b(t) − u2 (t),

(38)

where {Xt } and {b(t)} are the price Markov chains of the stock and bond prices at time t, respectively. According to Black–Scholes model, we have: dXt = Xt {Rdt + σ dBt }, db(t) = b(t)rdt ,

(39)

where Bt is classical Brownian motion. Also, r and R are constant interest rates of bond and stock for 0 < r < R, respectively. Please cite this article as: S. Yaghobipour and M. Yarahmadi, Solving quantum stochastic LQR optimal control problem in Fock space and its application in finance, Computers and Mathematics with Applications (2019), https://doi.org/10.1016/j.camwa.2019.12.016.

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In the next subsection, according to the presented algorithm, Merton portfolio allocation problem is solved. 5.2. Initial quantum space prescribing a. Describing the initial Hilbert space Consider a special stock market in which mathematical relations are modeled under a limit rule: the rate of return of investing, in a trading day, cannot exceed a fixed limit ±q%. Therefore, Ω = {−q, . . . , 0, . . . , q} is set of all possible changes in the stock price in the market. According to the algorithm of the proposed method, for describing the initial Hilbert space of the quantum system state corresponding to the financial system, each element of set Ω is q corresponded to a quantum value by function ψ : Ω → C. Also, an orthonormal set of the {|δn ⟩}n=−q is defined as follows [31,32]

δn : Ω → C, δn (k) =

1, 0,

k = n, k ̸ = n.

{

(40)

It is clear that the function δn defines the components of |δn ⟩. Thereby, the following vectors can be defined:

|ψ⟩ =

q ∑

ψ (n)|δn ⟩,

⟨ψ| =

n=−q

q ∑

ψ (n)⟨δn |,

(41)

n=−q

where ⟨δn |ψ⟩ = ψ (n). Now, vector |ψ⟩ is normalized as follows:

Ψ : Ω → C , Ψ (n) = √

1

⟨ψ|ψ⟩

ψ (n)

(42)

Therefore, the space H of all the functions |Ψ ⟩ is a Hilbert space of the quantum state of the quantum financial system, under the following scalar product:

⟨ψ, ϕ⟩ =

q ∑

ψ (n)ϕ (n) .

(43)

n=−q

In this stock market, the Hermitian operator of return is defined as follows [33]:

ˆ R : H → H, ˆ R=

q ∑

n|δn ⟩⟨δn |.

(44)

n=−q

In the other words:

{ ˆ R : H → H, ˆ R|Ψ (n)⟩ = n|Ψ (n)⟩.

(45)

The function |δn ⟩ is an eigen function of ˆ R corresponding to the eigenvalue n. Namely, ˆ R|δn ⟩ = n|δn ⟩. In quantum theory, the Schrödinger equation which describes the evolution of micro-world is considered as the corresponding dynamics and evolution equation [31]. Therefore, the following Schrödinger differential equation is considered as the rate of return evolution equation: i

∂Ψ = Hˆ Ψ , ∂t

(46)

where, the function Ψ : Ω × R → C describes the time-evolution of the presented financial system state and the ˆ is a Hermitian operator as: Hamiltonian H

ˆ = H

1 2µ

τˆ 2 + β Rˆ cos(ωt),

(47)

where µ and β are two positive constants. Also,

ˆ †, τˆ = F RF

F =

q ∑

2k/2 φ (2k/2 t − n)|δk ⟩⟨δn |,

k,n=−q

where k and n are scaling and translation parameters of orthonormal wavelet functions φk ,n , respectively [33]. b. Design the quantum stochastic LQR optimal control problem Now, for Merton portfolio allocation problem, the operator Xt = |Ψt ⟩⟨Ψt | is defined as the operator-valued variable of stock price. Also, the stochastic process of stock price {Xt } is replaced by the quantum stochastic flow (or the quantum version of a Markov chain) {jt (X )|t ⩾ 0} satisfied in Eqs. (8) and (9) for L = 0, S = 1 [28]. Therefore, as in the classical case, dynamics of wealth value for operator valued portfolio (u1 (t), α (t)), t ∈ [0, T ] is described as: dwt = u1 (t)djt (X ) + [1 − u1 (t)]db(t) − u2 (t)Idt ,

(48)

where, db = rbIdt and I is the identity matrix that has the same dimension as matrix jt (X ). Please cite this article as: S. Yaghobipour and M. Yarahmadi, Solving quantum stochastic LQR optimal control problem in Fock space and its application in finance, Computers and Mathematics with Applications (2019), https://doi.org/10.1016/j.camwa.2019.12.016.

S. Yaghobipour and M. Yarahmadi / Computers and Mathematics with Applications xxx (xxxx) xxx

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According to the assumptions mentioned above, the main objective is: min J(u2 ) = E u2

[∫

T

] [⟨X (s)h, Qs X (s)h⟩ + ⟨u2 (s)h, Rs u2 (s)h⟩]ds .

(49)

0

Therefore, the quantum stochastic optimal control problem in the symmetric Fock space is formulated as follows: minu2 ∈A J(u2 ) = E

[∫ T

] [⟨ X (s)h , Q X (s)h ⟩ + ⟨ u (s)h , R u (s)h ⟩] ds , s 2 s 2 0

subject to : dwt = u1 (t)djt (X ) + [1 − u1 (t)]db(t) − u2 (t)Idt , ˆ , X ])dt , djt (X ) = Rjt (i[H db = rbIdt , w(0) = w0 ,

(50)

ˆ is defined in Eq. (47). where Hermitian operator H 5.3. Input of the financial LQR 1. The quantum state |Ψ ⟩ is computed by (45) and (46), for the following quantum parameters: q = 10, β = 0.1, µ = 1 and the wavelet φ is Haar wavelet function. Consider the interest rates parameters as r = 0.05 and R = 0.11. 2. Initial bond price b(0) = 100$ and initial price of stock X (0) = 100$. 3. Let, Rt = tI21 for t ∈ (0, 100] and Qt = tI21 in the Riccati differential equation (51) with initial value R(0) = P(0) = Q (0) = I21 . 4. Consider the initial capital w (0) = 100,000$. 5.4. Steps of algorithm in simulation Step.1 According to problem formulation and problem parameters the following Riccati differential equation will be solved : 1 ˙ − P(t)R− P(t) t P(t) + Qt = 0.

(51)

Step.2 Quantum Stochastic LQR Optimal Control Rule Now according to the theorem 1, by considering the value function V (t , Xt ) = minu2 ∈A J(u2 ), the HJB equation corresponding to the quadratic performance (49) is:

(

)

ˆ , X ])) − Vw u2 (t)I Vt + Vw rP0 I − u1 (t)(rP0 I − Rjt (i[H +⟨Xt h, Qt Xt h⟩ + ⟨u2 h, Rt u2 h⟩ = 0.

(52)

By placing

[

]−1

ˆ , X ]) u∗1 (t) = rP0 rP0 I − Rjt (i[H

,

(53)

for the system described by dynamics dwt = u∗1 (t)djt (X ) + [1 − u∗1 (t)]dP0 (t) − u2 (t)Idt , the corresponding HJB equation is: Vt − Vw u2 (t)I + ⟨Xt h, Qt Xt h⟩ + ⟨u2 hRt u2 h⟩ = 0. Now, according to theorem 2, the operator valued optimal control process u∗2 is: u∗2 (t) = R−1 P(t)wt ,

(54)

where P(t) is the solution of the Riccati differential equation (51). Therefore, according to Eqs. (53) and (54) the values of strategy process u∗ (t) = [u∗1 (t), u∗2 (t)]T are computed. Step.3 Finally, according to the previous steps, the response of control process can be computed by Eq. (46). Please cite this article as: S. Yaghobipour and M. Yarahmadi, Solving quantum stochastic LQR optimal control problem in Fock space and its application in finance, Computers and Mathematics with Applications (2019), https://doi.org/10.1016/j.camwa.2019.12.016.

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S. Yaghobipour and M. Yarahmadi / Computers and Mathematics with Applications xxx (xxxx) xxx Table 2 Invested part in stock and portfolio return. Change-price

Invested part in stock (u1 )

Portfolio return

−10% −9% −8% −7% −6% −5% −4% −3% −2% −1%

28% 35% 48% 27% 33% 75% 47% 100% 73% 100% −4.3% −8.5% −1.16% −9.9% −6.2% −4.2% −5.6% −3.2% −2.5% −3.9% −2.9%

267% 244% 224% 188% 245% 224% 110% 189% 187% 117% −126% −15% −27% −59% −64% −65% −71% −76% −85% −91% −120%

0% 1% 2% 3% 4% 5% 6% 7% 8% 9% 10%

Fig. 1. The satisfaction value of consumption capital, u2 .

5.5. Simulation results In this subsection, the quantum stochastic LQR optimal control problem (quantum Merton problem) in the symmetric Fock space (50) is simulated for determined parameters and initial input values. The simulation results are shown in Table 2 and Figs. 1 and 2. Note that, for every operator X , jt (X ) = Ut∗ X ⊗ 1Ut where Ut is a unitary matrix. Therefore, according to Schur decomposition [34], jt (X ) is a diagonal matrix, i.e jt (X ) = diag(a1 , . . . , a2 1) where ai , i = 1, . . . , 21 are positive eigenvalues of X ⊗ 1. In Table 2, for each possible change value in stock price, an invested part in stock is determined by the proposed quantum stochastic optimal control method. For example, if the change-price of stock is −10% then by investing 28% of capital in stock, the portfolio return 267% is obtained. The negative values on second column means that trader must take position ‘‘sell’’ or does not invest in stock (risky asset). Additionally, the proposed Stochastic LQR Optimal Control algorithm can act simultaneously on the super∑Quantum q position of states (i.e. |ψ⟩ = n=−q ψ (n)|δn ⟩). Also, analogously to the classical computing model, the quantum algorithm consists of the quantum operation Xt = |Ψt ⟩⟨Ψt |, therefore the quantum algorithm could be parallelized. Therefore, in quantum formalism, trading position for all of the next events of the system is determined, simultaneously. Fig. 1 shows the consumption value of capital at any time t. Since, the satisfaction value of investing is evaluated by consuming capital, i.e. the smaller consumption value the more satisfaction value. Fig. 1 shows that by implementing operator valued strategy (u1 (t), α (t)), t ∈ [0, 100], the consumption values (cost of investing)u2 (t) decrease, significantly. Therefore, the investing in portfolio (u1 (t), α (t)) is a capable investment. Please cite this article as: S. Yaghobipour and M. Yarahmadi, Solving quantum stochastic LQR optimal control problem in Fock space and its application in finance, Computers and Mathematics with Applications (2019), https://doi.org/10.1016/j.camwa.2019.12.016.

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Fig. 2. The wealth evolution by portfolio (u1 (t), α (t)) in all situations of stock market.

Fig. 2 shows the wealth values obtained from investing in portfolio (u1 (t), α (t)) in all situations of stock market. Fig. 2 shows that if the stock price increases then trader should take position ‘‘sell’’ at the highest price and otherwise portfolio return will be negative at the future. 6. Conclusion In this paper, the stochastic LQR optimal problem in Fock space was investigated in quantum formalism, based on the quantum stochastic calculus. For this purpose, system dynamics was described by Hudson–Parthasarathy type Quantum Stochastic Differential Equation (QSDE) in Fock space and by associating a quadratic performance criterion with the QSDE, the Quantum Stochastic Linear Quadratic Regulator (QS-LQR) control problem was modeled and then solved by considering the new corresponding Hamilton–Jacobi–Bellman equation. Therefore, the operator-valued adapted control and state (response) processes were obtained based on the corresponding Riccati equation. For theoretical facilitating and implementing the method for solving the QS-LQR optimal control problem, two theorems were proved. Also, for indicating the advantages of the proposed method, a financial problem, Merton portfolio allocation problem was solved and simulated. The simulation results showed that portfolio optimal performances, minimum risk and maximum return, were achieved via presented method, in any situation of stock market. Also, the simulation results showed that in the quantum formalism for all of situations of stock market, the optimal strategy was determined and portfolio return was predicted, simultaneously. Finally, the parallel computing property of the quantum algorithm based on superposition property of the quantum systems is demonstrated in simulation of the presented method. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17]

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Please cite this article as: S. Yaghobipour and M. Yarahmadi, Solving quantum stochastic LQR optimal control problem in Fock space and its application in finance, Computers and Mathematics with Applications (2019), https://doi.org/10.1016/j.camwa.2019.12.016.

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Further reading [1] S. Mudchanatongsuk, J.A. Primbs, W. Wong, Optimal pairs trading: a stochastic control approach, Conference Location, IEEE, Seattle, WA, USA, 2008, pp. 1035–1039.

Please cite this article as: S. Yaghobipour and M. Yarahmadi, Solving quantum stochastic LQR optimal control problem in Fock space and its application in finance, Computers and Mathematics with Applications (2019), https://doi.org/10.1016/j.camwa.2019.12.016.