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Author Contreras, M.; Pena, J.P. pdf  doi
openurl 
  Title The quantum dark side of the optimal control theory Type Journal Article
  Year 2019 Publication Physica A-Statistical Mechanics And Its Applications Abbreviated Journal Physica A  
  Volume 515 Issue Pages 450-473  
  Keywords Optimal control theory; Pontryagin's equations; Hamilton-Jacobi-Bellman 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 Hamilton-Jacobi-Bellman 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 Hamilton-Jacobi-Bellman equation, which implies that the cost functional J(+)(x, t) must necessarily satisfy the usual Hamilton-Jacobi-Bellman 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 0378-4371 ISBN Medium  
  Area Expedition Conference  
  Notes WOS:000452941100042 Approved no  
  Call Number UAI @ eduardo.moreno @ Serial 952  
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Author Contreras, M.; Pellicer, R.; Villena, M. pdf  doi
openurl 
  Title Dynamic optimization and its relation to classical and quantum constrained systems Type Journal Article
  Year 2017 Publication Physica A-Statistical Mechanics And Its Applications Abbreviated Journal Physica A  
  Volume 479 Issue Pages 12-25  
  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 second-class 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 closed-loop lambda-strategy, the optimality condition for the action gives a consistency relation, which is associated to the Hamilton-Jacobi-Bellman 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 non-linear partial equation is obtained for the S function. For the right-hand side quantization, this is the Hamilton-Jacobi-Bellman equation, when S(x, t) is identified with the optimal value function. Thus, the Hamilton-Jacobi-Bellman 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 0378-4371 ISBN Medium  
  Area Expedition Conference  
  Notes WOS:000400213800002 Approved no  
  Call Number UAI @ eduardo.moreno @ Serial 731  
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Author Mahajan, S.M.; Asenjo, F.A. pdf  doi
openurl 
  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 1435-1449  
  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, Klein-Gordon, and Pauli-Schrodinger and Feynman-GellMaan 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 energy-density 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 0020-7748 ISBN Medium  
  Area Expedition Conference  
  Notes WOS:000352858600004 Approved no  
  Call Number UAI @ eduardo.moreno @ Serial 485  
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Author Asenjo, F.A.; Mahajan, S.M. pdf  doi
openurl 
  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; Feynman-GellMann equation  
  Abstract We explore the fluid version of the quadratic form of the Dirac equation, sometimes called the Feynman-Gell-Mann 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 non-relativistic 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 0031-8949 ISBN Medium  
  Area Expedition Conference  
  Notes WOS:000349301500001 Approved no  
  Call Number UAI @ eduardo.moreno @ Serial 458  
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Author Hojman, S.A. pdf  url
openurl 
  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 336-339  
  Keywords Quantum mechanics; modified Dirac-Kronig-Penney 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 0035-001x ISBN Medium  
  Area Expedition Conference  
  Notes WOS:000341802200001 Approved no  
  Call Number UAI @ eduardo.moreno @ Serial 409  
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Author Contreras, M.; Hojman, S.A. pdf  doi
openurl 
  Title Option pricing, stochastic volatility, singular dynamics and constrained path integrals Type Journal Article
  Year 2014 Publication Physica A-Statistical Mechanics And Its Applications Abbreviated Journal Physica A  
  Volume 393 Issue Pages 391-403  
  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 0378-4371 ISBN Medium  
  Area Expedition Conference  
  Notes WOS:000328179200034 Approved no  
  Call Number UAI @ eduardo.moreno @ Serial 335  
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Author Hojman, S.A.; Gamboa, J.; Mendez, F. pdf  doi
openurl 
  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; non-commutative geometry  
  Abstract The inverse problem of calculus of variations and s-equivalence are re-examined by using results obtained from non-commutative 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 s-equivalent systems may be non-equivalent 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 0217-7323 ISBN Medium  
  Area Expedition Conference  
  Notes WOS:000310278700003 Approved no  
  Call Number UAI @ eduardo.moreno @ Serial 249  
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Author Contreras, M.; Pellicer, R.; Villena, M.; Ruiz, A. pdf  doi
openurl 
  Title A quantum model of option pricing: When Black-Scholes meets Schrodinger and its semi-classical limit Type Journal Article
  Year 2010 Publication Physica A-Statistical Mechanics And Its Applications Abbreviated Journal Physica A  
  Volume 389 Issue 23 Pages 5447-5459  
  Keywords Black-Scholes model; Arbitrage; Option pricing; Quantum mechanics; Semi-classical methods  
  Abstract The Black-Scholes 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, short-term volatility, extreme discontinuities, or serial correlations; the classical non-arbitrage assumption of the Black-Scholes model is violated, implying a non-risk-free portfolio. From Haven (2002) [1] we know that an arbitrage environment is a necessary condition to embedding the Black-Scholes option pricing model in a more general quantum physics setting. The aim of this paper is to propose a new Black-Scholes-Schrodinger 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 Black-Scholes 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 Black-Scholes 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 0378-4371 ISBN Medium  
  Area Expedition Conference  
  Notes WOS:000283904000012 Approved no  
  Call Number UAI @ eduardo.moreno @ Serial 116  
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