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Author 
Contreras, M.; Pena, J.P. 


Title 
The quantum dark side of the optimal control theory 
Type 
Journal Article 

Year 
2019 
Publication 
Physica AStatistical Mechanics And Its Applications 
Abbreviated Journal 
Physica A 


Volume 
515 
Issue 

Pages 
450473 


Keywords 
Optimal control theory; Pontryagin's equations; HamiltonJacobiBellman equation; Constrained systems; Dirac's method; Quantum mechanics 


Abstract 
In a recent article, a generic optimal control problem was studied from a physicist's point of view (Contreras et al. 2017). Through this optic, the Pontryagin equations are equivalent to the Hamilton equations of a classical constrained system. By quantizing this constrained system, using the right ordering of the operators, the corresponding quantum dynamics given by the Schrodinger equation is equivalent to that given by the HamiltonJacobiBellman equation of Bellman's theory. The conclusion drawn there were based on certain analogies between the equations of motion of both theories. In this paper, a closer and more detailed examination of the quantization problem is carried out, by considering three possible quantization procedures: right quantization, left quantization, and Feynman's path integral approach. The Bellman theory turns out to be the classical limit h > 0 of these three different quantum theories. Also, the exact relation of the phase S(x, t) of the wave function Psi(x, t) = e(i/hS(x,t)) of the quantum theory with Bellman's cost function J(+)(x, t) is obtained. In fact, S(x, t) satisfies a 'conjugate' form of the HamiltonJacobiBellman equation, which implies that the cost functional J(+)(x, t) must necessarily satisfy the usual HamiltonJacobiBellman equation. Thus, the Bellman theory effectively corresponds to a quantum view of the optimal control problem. (C) 2018 Elsevier B.V. All rights reserved. 


Address 
[Contreras G, Mauricio] Univ Adolfo Ibanez, Fac Ingn & Ciencias, Santiago, Chile, Email: mauricio.contreras@uai.cl 


Corporate Author 

Thesis 



Publisher 
Elsevier Science Bv 
Place of Publication 

Editor 



Language 
English 
Summary Language 

Original Title 



Series Editor 

Series Title 

Abbreviated Series Title 



Series Volume 

Series Issue 

Edition 



ISSN 
03784371 
ISBN 

Medium 



Area 

Expedition 

Conference 



Notes 
WOS:000452941100042 
Approved 
no 


Call Number 
UAI @ eduardo.moreno @ 
Serial 
952 

Permanent link to this record 




Author 
Contreras, M.; Pellicer, R.; Villena, M. 


Title 
Dynamic optimization and its relation to classical and quantum constrained systems 
Type 
Journal Article 

Year 
2017 
Publication 
Physica AStatistical Mechanics And Its Applications 
Abbreviated Journal 
Physica A 


Volume 
479 
Issue 

Pages 
1225 


Keywords 
Dynamic optimization; Constrained systems; Dirac's method; Quantum mechanics 


Abstract 
We study the structure of a simple dynamic optimization problem consisting of one state and one control variable, from a physicist's point of view. By using an analogy to a physical model, we study this system in the classical and quantum frameworks. Classically, the dynamic optimization problem is equivalent to a classical mechanics constrained system, so we must use the Dirac method to analyze it in a correct way. We find that there are two secondclass constraints in the model: one fix the momenta associated with the control variables, and the other is a reminder of the optimal control law. The dynamic evolution of this constrained system is given by the Dirac's bracket of the canonical variables with the Hamiltonian. This dynamic results to be identical to the unconstrained one given by the Pontryagin equations, which are the correct classical equations of motion for our physical optimization problem. In the same Pontryagin scheme, by imposing a closedloop lambdastrategy, the optimality condition for the action gives a consistency relation, which is associated to the HamiltonJacobiBellman equation of the dynamic programming method. A similar result is achieved by quantizing the classical model. By setting the wave function Psi (x, t) = e(is(x,t)) in the quantum Schrodinger equation, a nonlinear partial equation is obtained for the S function. For the righthand side quantization, this is the HamiltonJacobiBellman equation, when S(x, t) is identified with the optimal value function. Thus, the HamiltonJacobiBellman equation in Bellman's maximum principle, can be interpreted as the quantum approach of the optimization problem. (C) 2017 Elsevier B.V. All rights reserved. 


Address 
[Contreras, Mauricio; Pellicer, Rely; Villena, Marcelo] Univ Adolfo Ibanez, Fac Engn & Sci, Santiago, Region Metropol, Chile, Email: mauricio.contreras@uai.cl 


Corporate Author 

Thesis 



Publisher 
Elsevier Science Bv 
Place of Publication 

Editor 



Language 
English 
Summary Language 

Original Title 



Series Editor 

Series Title 

Abbreviated Series Title 



Series Volume 

Series Issue 

Edition 



ISSN 
03784371 
ISBN 

Medium 



Area 

Expedition 

Conference 



Notes 
WOS:000400213800002 
Approved 
no 


Call Number 
UAI @ eduardo.moreno @ 
Serial 
731 

Permanent link to this record 




Author 
Mahajan, S.M.; Asenjo, F.A. 


Title 
Hot Fluids and Nonlinear Quantum Mechanics 
Type 
Journal Article 

Year 
2015 
Publication 
International Journal Of Theoretical Physics 
Abbreviated Journal 
Int. J. Theor. Phys. 


Volume 
54 
Issue 
5 
Pages 
14351449 


Keywords 
Nonlinear quantum mechanics; Fluids; Temperature; High energy density physics 


Abstract 
A hot relativistic fluid is viewed as a collection of quantum objects that represent interacting elementary particles. We present a conceptual framework for deriving nonlinear equations of motion obeyed by these hypothesized objects. A uniform phenomenological prescription, to affect the quantum transition from a corresponding classical system, is invoked to derive the nonlinear Schrodinger, KleinGordon, and PauliSchrodinger and FeynmanGellMaan equations. It is expected that the emergent hypothetical nonlinear quantum mechanics would advance, in a fundamental way, both the conceptual understanding and computational abilities, particularly, in the field of extremely high energydensity physics. 


Address 
[Mahajan, Swadesh M.] Univ Texas Austin, Inst Fus Studies, Austin, TX 78712 USA, Email: mahajan@mail.utexas.edu; 


Corporate Author 

Thesis 



Publisher 
Springer/Plenum Publishers 
Place of Publication 

Editor 



Language 
English 
Summary Language 

Original Title 



Series Editor 

Series Title 

Abbreviated Series Title 



Series Volume 

Series Issue 

Edition 



ISSN 
00207748 
ISBN 

Medium 



Area 

Expedition 

Conference 



Notes 
WOS:000352858600004 
Approved 
no 


Call Number 
UAI @ eduardo.moreno @ 
Serial 
485 

Permanent link to this record 




Author 
Asenjo, F.A.; Mahajan, S.M. 


Title 
Relativistic quantum vorticity of the quadratic form of the Dirac equation 
Type 
Journal Article 

Year 
2015 
Publication 
Physica Scripta 
Abbreviated Journal 
Phys. Scr. 


Volume 
90 
Issue 
1 
Pages 
4 pp 


Keywords 
relativistic quantum mechanics; hydrodynamical version; FeynmanGellMann equation 


Abstract 
We explore the fluid version of the quadratic form of the Dirac equation, sometimes called the FeynmanGellMann equation. The dynamics of the quantum spinor field is represented by equations of motion for the fluid density, the velocity field, and the spin field. In analogy with classical relativistic and nonrelativistic quantum theories, the fully relativistic fluid formulation of this equation allows a vortex dynamics. The vortical form is described by a total tensor field that is the weighted combination of the inertial, electromagnetic and quantum forces. The dynamics contrives the quadratic form of the Dirac equation as a total vorticity free system. 


Address 
[Asenjo, Felipe A.] Univ Adolfo Ibanez, Fac Ingn & Ciencias, Santiago, Chile, Email: felipe.asenjo@uai.cl 


Corporate Author 

Thesis 



Publisher 
Iop Publishing Ltd 
Place of Publication 

Editor 



Language 
English 
Summary Language 

Original Title 



Series Editor 

Series Title 

Abbreviated Series Title 



Series Volume 

Series Issue 

Edition 



ISSN 
00318949 
ISBN 

Medium 



Area 

Expedition 

Conference 



Notes 
WOS:000349301500001 
Approved 
no 


Call Number 
UAI @ eduardo.moreno @ 
Serial 
458 

Permanent link to this record 




Author 
Hojman, S.A. 


Title 
Origin of conical dispersion relations 
Type 
Journal Article 

Year 
2014 
Publication 
Revista Mexicana De Fisica 
Abbreviated Journal 
Rev. Mex. Fis. 


Volume 
60 
Issue 
5 
Pages 
336339 


Keywords 
Quantum mechanics; modified DiracKronigPenney potential; conical dispersion relations 


Abstract 
A mechanism that produces conical dispersion relations is presented. A Kronig Penney one dimensional array with two different strengths delta function potentials gives rise to both the gap closure and the dispersion relation observed in graphene and other materials. The Schrodinger eigenvalue problem is locally invariant under, the infinite dimensional Virasoro algebra near conical dispersion points in reciprocal space, thus suggesting a possible relation to string theory. 


Address 
[Hojman, Sergio A.] Univ Adolfo Ibanez, Dept Ciencias, Fac Artes Liberales, Fac Ingn & Ciencias, Santiago, Chile, Email: sergio.hojman@uai.cl 


Corporate Author 

Thesis 



Publisher 
Soc Mexicana Fisica 
Place of Publication 

Editor 



Language 
English 
Summary Language 

Original Title 



Series Editor 

Series Title 

Abbreviated Series Title 



Series Volume 

Series Issue 

Edition 



ISSN 
0035001x 
ISBN 

Medium 



Area 

Expedition 

Conference 



Notes 
WOS:000341802200001 
Approved 
no 


Call Number 
UAI @ eduardo.moreno @ 
Serial 
409 

Permanent link to this record 




Author 
Contreras, M.; Hojman, S.A. 


Title 
Option pricing, stochastic volatility, singular dynamics and constrained path integrals 
Type 
Journal Article 

Year 
2014 
Publication 
Physica AStatistical Mechanics And Its Applications 
Abbreviated Journal 
Physica A 


Volume 
393 
Issue 

Pages 
391403 


Keywords 
Option pricing; Stochastic volatility; Quantum mechanics; Singular Lagrangian systems; Dirac's method; Constrained Hamiltonian path integrals 


Abstract 
Stochastic volatility models have been widely studied and used in the financial world. The Heston model (Heston, 1993) [7] is one of the best known models to deal with this issue. These stochastic volatility models are characterized by the fact that they explicitly depend on a correlation parameter p which relates the two Brownian motions that drive the stochastic dynamics associated to the volatility and the underlying asset. Solutions to the Heston model in the context of option pricing, using a path integral approach, are found in Lemmens et al. (2008) [21] while in Baaquie (2007,1997) [12,13] propagators for different stochastic volatility models are constructed. In all previous cases, the propagator is not defined for extreme cases rho = +/ 1. It is therefore necessary to obtain a solution for these extreme cases and also to understand the origin of the divergence of the propagator. In this paper we study in detail a general class of stochastic volatility models for extreme values rho = +/ 1 and show that in these two cases, the associated classical dynamics corresponds to a system with second class constraints, which must be dealt with using Dirac's method for constrained systems (Dirac, 1958,1967) [22,23] in order to properly obtain the propagator in the form of a Euclidean Hamiltonian path integral (Henneaux and Teitelboim, 1992) [25]. After integrating over momenta, one gets an Euclidean Lagrangian path integral without constraints, which in the case of the Heston model corresponds to a path integral of a repulsive radial harmonic oscillator. In all the cases studied, the price of the underlying asset is completely determined by one of the second class constraints in terms of volatility and plays no active role in the path integral. (C) 2013 Elsevier B.V. All rights reserved. 


Address 
[Contreras, Mauricio] Univ Adolfo Ibanez, Fac Ingn & Ciencias, Santiago, Chile, Email: mauricio.contreras@uai.cl 


Corporate Author 

Thesis 



Publisher 
Elsevier Science Bv 
Place of Publication 

Editor 



Language 
English 
Summary Language 

Original Title 



Series Editor 

Series Title 

Abbreviated Series Title 



Series Volume 

Series Issue 

Edition 



ISSN 
03784371 
ISBN 

Medium 



Area 

Expedition 

Conference 



Notes 
WOS:000328179200034 
Approved 
no 


Call Number 
UAI @ eduardo.moreno @ 
Serial 
335 

Permanent link to this record 




Author 
Hojman, S.A.; Gamboa, J.; Mendez, F. 


Title 
Dynamics Determines Geometry 
Type 
Journal Article 

Year 
2012 
Publication 
Modern Physics Letters A 
Abbreviated Journal 
Mod. Phys. Lett. A 


Volume 
27 
Issue 
33 
Pages 
14 pp 


Keywords 
Classical and quantum mechanics; noncommutative geometry 


Abstract 
The inverse problem of calculus of variations and sequivalence are reexamined by using results obtained from noncommutative geometry ideas. The role played by the structure of the modified Poisson brackets is discussed in a general context and it is argued that classical sequivalent systems may be nonequivalent at the quantum mechanical level. This last fact is explicitly discussed comparing different approaches to deal with the NairPolychronakos oscillator. 


Address 
[Gamboa, J.; Mendez, F.] Univ Santiago Chile, Dept Fis, Santiago, Chile, Email: sergio.hojman@uai.cl; 


Corporate Author 

Thesis 



Publisher 
World Scientific Publ Co Pte Ltd 
Place of Publication 

Editor 



Language 
English 
Summary Language 

Original Title 



Series Editor 

Series Title 

Abbreviated Series Title 



Series Volume 

Series Issue 

Edition 



ISSN 
02177323 
ISBN 

Medium 



Area 

Expedition 

Conference 



Notes 
WOS:000310278700003 
Approved 
no 


Call Number 
UAI @ eduardo.moreno @ 
Serial 
249 

Permanent link to this record 




Author 
Contreras, M.; Pellicer, R.; Villena, M.; Ruiz, A. 


Title 
A quantum model of option pricing: When BlackScholes meets Schrodinger and its semiclassical limit 
Type 
Journal Article 

Year 
2010 
Publication 
Physica AStatistical Mechanics And Its Applications 
Abbreviated Journal 
Physica A 


Volume 
389 
Issue 
23 
Pages 
54475459 


Keywords 
BlackScholes model; Arbitrage; Option pricing; Quantum mechanics; Semiclassical methods 


Abstract 
The BlackScholes equation can be interpreted from the point of view of quantum mechanics, as the imaginary time Schrodinger equation of a free particle. When deviations of this state of equilibrium are considered, as a product of some market imperfection, such as: Transaction cost, asymmetric information issues, shortterm volatility, extreme discontinuities, or serial correlations; the classical nonarbitrage assumption of the BlackScholes model is violated, implying a nonriskfree portfolio. From Haven (2002) [1] we know that an arbitrage environment is a necessary condition to embedding the BlackScholes option pricing model in a more general quantum physics setting. The aim of this paper is to propose a new BlackScholesSchrodinger model based on the endogenous arbitrage option pricing formulation introduced by Contreras et al. (2010) [2]. Hence, we derive a more general quantum model of option pricing, that incorporates arbitrage as an external time dependent force, which has an associated potential related to the random dynamic of the underlying asset price. This new resultant model can be interpreted as a Schrodinger equation in imaginary time for a particle of mass 1/sigma(2) with a wave function in an external field force generated by the arbitrage potential. As pointed out above, this new model can be seen as a more general formulation, where the perfect market equilibrium state postulated by the BlackScholes model represent a particular case. Finally, since the Schrodinger equation is in place, we can apply semiclassical methods, of common use in theoretical physics, to find an approximate analytical solution of the BlackScholes equation in the presence of market imperfections, as it is the case of an arbitrage bubble. Here, as a numerical illustration of the potential of this Schrodinger equation analogy, the semiclassical approximation is performed for different arbitrage bubble forms (step, linear and parabolic) and compare with the exact solution of our general quantum model of option pricing. (C) 2010 Elsevier B.V. All rights reserved. 


Address 
[Contreras, Mauricio; Pellicer, Rely; Villena, Marcelo; Ruiz, Aaron] Adolfo Ibanez Univ, Fac Sci & Engn, Santiago, Chile, Email: mauricio.contreras@uai.cl 


Corporate Author 

Thesis 



Publisher 
Elsevier Science Bv 
Place of Publication 

Editor 



Language 
English 
Summary Language 

Original Title 



Series Editor 

Series Title 

Abbreviated Series Title 



Series Volume 

Series Issue 

Edition 



ISSN 
03784371 
ISBN 

Medium 



Area 

Expedition 

Conference 



Notes 
WOS:000283904000012 
Approved 
no 


Call Number 
UAI @ eduardo.moreno @ 
Serial 
116 

Permanent link to this record 