Quantum Stochastic Differential Research Articles

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

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.

Optimal control design for a class of quantum stochastic systems with financial applications

The purpose of this paper is to design an optimal quantum controller for a class of stochastic systems with application in financial problems. Dynamics of the system is prescribed via a Quantum Stochastic Differential System (QSDES) with a quantum Brownian motion on a quantum probability space. A theorem for guaranteeing the existence and uniqueness of solutions to the QSDES is proved. Additionally, a new optimal stochastic control problem is formulated and based on the necessary optimality conditions, an optimal quantum control law is designed, explicitly. Four theorems and two lemmas, for facilitating the optimal controller design algorithm, are proved. Finally, for demonstrating the applicable results, two financial problems, Merton portfolio allocation and optimal pairs trading problem are simulated by using the presented method. As the simulation results indicate, portfolio optimal performances, minimum risk and maximum return, are achieved via presented method.

37th International Conference on Quantum Probability and Related Topics (QP37)

ForewordThe present volume of Journal of Physics: Conference Series contains the Proceedings of The 37-th International Conference on Quantum Probability and Related Topics (QP37), held in the Faculty of Science of the International Islamic University Malaysia, Kuantan, Malaysia, in August 22-26, 2016.The intrinsic multi-disciplinary character of Quantum Probability is emphasized by its fruitful interaction with different aspects of contemporary mathematics. In addition to the traditional topics of Quantum Probability such as the recently discovered bridge with the classical theory of Orthogonal Polynomials, Quantum Field Theory, Quantum statistical Mechanics, the theory of Quantum Markov semigroups and of Quantum Markov chains and fields, Quantum Stochastic Differential Equations, operator algebras and dynamical systems, the conference has stimulated active communications with the theory of Quadratic operators, both in its classical and quantum aspects, which has a stronghold in the Faculty of Science of International Islamic University Malaysia (Kuantan).The constructive interaction between pure and applied mathematics is another characteristic feature that has accompanied Quantum Probability in its almost fifty years of developments since the early 1970‘s. This is reflected, in the present volume, by a substantial presence of applications to several concrete models, not only of the mathematical techniques developed within Quantum Probability, but also in a multiplicity of other techniques which range from soliton theory, Fourier analysis, the theory of Boundary Value problems, ….The history of the 36 Quantum Probability conferences preceding the present one, illustrated by a collection of 33 proceedings volumes, shows that in several occasions these interactions between apparently far research directions have given rise to fruitful new points of view that have enriched all disciplines involved with new perspectives, new results and new problems.The program of conference included invited lectures (40 minutes) and poster section of more than 80 participants, including invited speakers from India, Italy, Japan, Jordan, Malaysia, The Republic of Korea, Singapore, Taiwan, Tunis, UK, USA and Uzbekistan. We would like to thank the Invited Speakers for their significant contributions to the conference. We would also like to thank the members of the International Scientific Committee and the members of the Organizing Committee. We cannot end without expressing our many thanks to our sponsors for their financial support. It is the strongest hope of the organizers that similar unexpected developments will be stimulated also by the present conference.Guest EditorsLuigi Accardi, Farrukh Mukhamedov, Pah Chin Hee

Quantum stochastic dynamics in multi-photon optics

Multi-photon models are theoretically and experimentally important because in them quantum properly phenomena are verified; as well as squeezed light and quantum entanglement also play a relevant role in quantum information and quantum communication (see Refs. 18–20).In this paper we study a generic model of a multi-photon system with an arbitrary number of pumping and subharmonics fields. This model includes measurement on the system, as could be direct or homodyne detection and we demonstrate the existence of dynamics in the context of Continuous Measurement Theory of Open Quantum Systems (see Refs. 1–11) using Quantum Stochastic Differential Equations with unbounded coefficients (see Refs. 10–15).

Simulation of Quantum Dynamics Based on the Quantum Stochastic Differential Equation

The quantum stochastic differential equation derived from the Lindblad form quantum master equation is investigated. The general formulation in terms of environment operators representing the quantum state diffusion is given. The numerical simulation algorithm of stochastic process of direct photodetection of a driven two-level system for the predictions of the dynamical behavior is proposed. The effectiveness and superiority of the algorithm are verified by the performance analysis of the accuracy and the computational cost in comparison with the classical Runge-Kutta algorithm.

On the existence of continuous selections of solution and reachable sets of quantum stochastic differential inclusions

We prove that the map that associates to the initial value the set of solutions to the Lipschitzian Quantum Stochastic Differential Inclusion (QSDI) admits a selection continuous from the locally convex space of stochastic processes to the adapted and weakly absolutely continuous space of solutions. As a corollary, we show that the reachable sets admit some continuous selections. In the framework of the Hudson - Parthasarathy formulations of quantum stochastic calculus, our results are achieved subject to some compactness conditions on the set of initial values and on some coefficients of the inclusion. The results here compliment similar results in our previous work in [3] where continuous selections defined on the set of the matrix elements of initial values were established. JONAMP Vol. 11 2007: pp. 71-82

Singular Perturbation of Quantum Stochastic Differential Equations with Coupling Through an Oscillator Mode

We consider a physical system which is coupled indirectly to a Markovian resevoir through an oscillator mode. This is the case, for example, in the usual model of an atomic sample in a leaky optical cavity which is ubiquitous in quantum optics. In the strong coupling limit the oscillator can be eliminated entirely from the model, leaving an effective direct coupling between the system an the resevoir. Here we provide a mathematically rigorous treatment of this limit as a weak limit of the time evolution and observables on a suitably chosen exponential domain in Fock space. The resulting effective model may contain emission and absorption as well as scattering interactions.

On the Generator of the Solution of a Quantum Stochastic Differential Equation

If the Hamiltonian of the system is a bounded selfadjoint operator, we give a simple alternative characterization of the infinitesimal generator iC of the unitary group associated with a Quantum Stochastic Differential Equation (QSDE). By using the defect indices of the symmetric operator C, we give a sufficient condition that assures that iC is the restriction of a generator of a strongly continuous semigroup of isometries in the case when the Hamiltonian of the system is an unbounded symmetric operator.

Exponential Formula for the Reachable Sets of Quantum Stochastic Differential Inclusions

We establish an exponential formula for the reachable sets of quantum stochastic differential inclusions (QSDI) which are locally Lipschitzian with convex values. Our main results partially rely on an auxilliary result concerning the density, in the topology of the locally convex space of solutions, of the set of trajectories whose matrix elements are continuously differentiable. By applying the exponential formula, we obtain results concerning convergence of the discrete approximations of the reachable set of the QSDI. This extends similar results of Wolenski[20] for classical differential inclusions to the present noncommutative quantum setting.

What Is a Quantum Stochastic Differential Equation from the Point of View of Functional Analysis

We prove that a quantum stochastic differential equation is the interaction representation of the Cauchy problem for the Schrodinger equation with Hamiltonian given by a certain operator restricted by a boundary condition. If the deficiency index of the boundary-value problem is trivial, then the corresponding quantum stochastic differential equation has a unique unitary solution. Therefore, by the deficiency index of a quantum stochastic differential equation we mean the deficiency index of the related symmetric boundary-value problem.

Lagrangian Quadrature Schemes for Computing Weak Solutions of Quantum Stochastic Differential Equations

Lagrangian quadrature schemes for computing weak solutions of Lipschitzian quantum stochastic differential equations are introduced and studied. This is accomplished within the framework of the Hudson--Parthasarathy formulation of quantum stochastic calculus and subject to matrix elements of solution being sufficiently differentiable. Results concerning convergence of these schemes in the topology of the locally convex space of solution are presented. Numerical examples are given.

On Convergence of One-Step Schemes for Weak Solutions of Quantum Stochastic Differential Equations

Several one-step schemes for computing weak solutions of Lipschitzian quantum stochastic differential equations (QSDE) driven by certain operator-valued stochastic processes associated with creation, annihilation and gauge operators of quantum field theory are introduced and studied. This is accomplished within the framework of the Hudson–Parthasarathy formulation of quantum stochastic calculus and subject to the matrix elements of solution being sufficiently differentiable. Results concerning convergence of these schemes in the topology of the locally convex space of solution are presented. It is shown that the Euler–Maruyama scheme,with respect to weak convergence criteria for Ito stochastic differential equation is a special case of Euler schemes in this framework. Numerical examples are given.

On Quantum Stochastic Differential Equations as Dirac Boundary-Value Problems

We prove that a single-jump unitary quantum stochastic evolution is unitarily equivalent to the Dirac boundary-value problem on the half-line in an extended space. It is shown that this solvable model can be derived from the Schrodinger boundary-value problem for a positive relativistic Hamiltonian on the half-line as the inductive ultrarelativistic limit corresponding to the input flow of Dirac particles with asymptotically infinite momenta. Thus the problem of stochastic approximation can be reduced to a quantum mechanical boundary-value problem in the extended space. The problem of microscopic time reversibility is also discussed in the paper.

Mild Solutions of Quantum Stochastic Differential Equations

We introduce the concept of a mild solution for the right Hudson-Parthasarathy quantum stochastic differential equation, prove existence and uniqueness results, and show the correspondence between our definition and similar ideas in the theory of classical stochastic differential equations. The conditions that a process must satisfy in order for it to be a mild solution are shown to be strictly weaker than those for it to be a strong solution by exhibiting a class of coefficient matrices for which a mild unitary solution can be found, but for which no strong solution exists.