Schrodinger Problem Research Articles

Solution for nonvariational quasilinear elliptic systems via sub-supersolution technique and Galerkin method

In this paper, we obtain the existence of positive solution for a system of quasilinear Schrodinger equations with concave nonlinearities which is related to several applications in Hydrodynamics, Heidelberg Ferromagnetism and Magnus Theory, Condensed Matter Theory, Dissipative Quantum Mechanics and nanotubes and fullerene-related structures. The quasilinear Schrodinger problem is studied by considering a suitable change of variables which transforms the original problem in to a semilinear one. By means of the several properties of the change of variables, constructions of suitable sub-supersolutions, monotonic iteration arguments and the Galerkin method, we obtain the existence of solution for the semilinear problem. The paper is divided in two parts. In the first one, we use the method of sub-supersolutions to obtain a solution for the problem. In the second part, we use the Galerkin method and a comparison argument to obtain a solution for the system considered. An important feature is that the sub-supersolution approach is rare in the literature for the type of problem considered here and the Galerkin method was not used to consider quasilinear Schrodinger equations.

Stochastic optimal transport revisited

We prove the Duality Theorems for the stochastic optimal transportation problems with a convex cost function without a regularity assumption that is often supposed in the proof of the lower semicontinuity of an action integral. In our new approach, we prove that the stochastic optimal transportation problems with a convex cost function are equivalent to a class of variational problems for the Fokker-Planck equation, which lets us revisit them. It is done by the so-called superposition principle and by an idea from the mather theory. The superposition principle is the construction of a semimartingale from the Fokker-Planck equation and can be considered a class of the so-called marginal problems that construct stochastic processes from given marginal distributions. It was first considered in stochastic mechanics by E. Nelson, called Nelson's problem, and was proved by E. Carlen first. The semimartingale is called the Nelson process, provided it is Markovian. We also consider the Markov property of a minimizer of the stochastic optimal transportation problem with a nonconvex cost in a one-dimensional case. In the proof, the superposition principle and the minimizer of an optimal transportation problem with a concave cost function play crucial roles. Lastly, we prove the semiconcavity and the Lipschitz continuity of Schrodinger's problem that is a typical example of the stochastic optimal transportation problem.

Dispersion-Constrained Martingale Schrödinger Problems and the Exact Joint S&P 500/VIX Smile Calibration Puzzle

We solve for the first time (*) a longstanding puzzle of quantitative finance that has often been described as the Holy Grail of volatility modeling: build a model that jointly and exactly calibrates to the prices of S&P 500 (SPX) options, VIX futures, and VIX options. So far the best attempts, which used parametric continuous-time (jump-)diffusion models on the SPX, only produced approximate fits. We use a very different, nonparametric and discrete-time approach. Given a VIX future maturity T1, we consider the set P of all joint probability measures on the SPX at T1 the VIX at T1, and the SPX at T2 = T1 + 30 days which are perfectly calibrated to the full SPX smiles at T1 and T2, and the full VIX smile at T1, and which also satisfy the martingality constraint on the SPX as well as the requirement that the VIX is the implied volatility of the 30-day log-contract on the SPX. We first consider robust hedging in this setting. By casting the superreplication problem as what we call a dispersion-constrained martingale optimal transport problem, we establish a strong duality theorem and, as a result, prove that the absence of joint SPX/VIX arbitrage is equivalent to the set P of jointly calibrating models being nonempty. Should they arise, joint arbitrages are identified using classical linear programming. In the absence of joint arbitrage, we then provide a solution to the joint calibration puzzle by solving what we call a dispersion-constrained martingale Schrodinger problem: we choose a reference measure and build the unique jointly calibrating model that minimizes the relative entropy. We establish several dual versions of the problem, one of which has a natural financial interpretation in terms of exponential utility indifference pricing, and prove absence of duality gaps. The minimum entropy jointly calibrating model is explicit in terms of what we call the dual Schrodinger portfolio, i.e., the maximizer of the dual problems, should it exist. We numerically compute this Schrodinger portfolio using an extension of the Sinkhorn algorithm, in the spirit of De March and Henry-Labordere (2019). Our numerical experiments show that the algorithm performs very well in both low and high volatility regimes. Along the way, we provide new variants, as well as a new proof, of strong duality theorems for the classical Schrodinger problem and for a mixed Schrodinger-Monge-Kantorovich problem (also known as entropic optimal transport problem) that has recently attracted a lot of attention in the optimal transport community, which are interesting in themselves. Our methodology applies not only to the VIX, but also to any index computed as a function of the price of an option on some underlying asset. (*) A first, much shorter version of this article was published in Risk in April 2020. This version includes new theorems, proofs, analysis, and numerical tests. In particular it develops a theory of dispersion-constrained martingale Schrodinger problems and proves strong duality results for them.

Stochastic Control Liaisons: Richard Sinkhorn Meets Gaspard Monge on a Schrödinger Bridge

In 1931--1932, Erwin Schrodinger studied a hot gas Gedankenexperiment (an instance of large deviations of the empirical distribution). Schrodinger's problem represents an early example of a fundame...

Multiplicative Schrödinger problem and the Dirichlet transport

We consider an optimal transport problem on the unit simplex whose solutions are given by gradients of exponentially concave functions and prove two main results. First, we show that the optimal transport is the large deviation limit of a particle system of Dirichlet processes transporting one probability measure on the unit simplex to another by coordinatewise multiplication and normalizing. The structure of our Lagrangian and the appearance of the Dirichlet process relate our problem closely to the entropic measure on the Wasserstein space as defined by von-Renesse and Sturm in the context of Wasserstein diffusion. The limiting procedure is a triangular limit where we allow simultaneously the number of particles to grow to infinity while the ‘noise’ tends to zero. The method, which generalizes easily to many other cost functions, including the squared Euclidean distance, provides a novel combination of the Schrodinger problem approach due to C. Leonard and the related Brownian particle systems by Adams et al. which does not require gamma convergence. Second, we analyze the behavior of entropy along the paths of transport. The reference measure on the simplex is taken to be the Dirichlet measure with all zero parameters which relates to the finite-dimensional distributions of the entropic measure. The interpolating curves are not the usual McCann lines. Nevertheless we show that entropy plus a multiple of the transport cost remains convex, which is reminiscent of the semiconvexity of entropy along lines of McCann interpolations in negative curvature spaces. We also obtain, under suitable conditions, dimension-free bounds of the optimal transport cost in terms of entropy.

Small Noise Limit and Convexity for Generalized Incompressible Flows, Schrödinger Problems, and Optimal Transport

This paper is concerned with six variational problems and their mutual connections: the quadratic Monge–Kantorovich optimal transport, the Schrodinger problem, Brenier’s relaxed model for incompressible fluids, the so-called Brodinger problem recently introduced by Arnaudon et al. (An entropic interpolation problem for incompressible viscid fluids, arXiv preprint arXiv:1704.02126, 2017) the multiphase Brenier model, and the multiphase Brodinger problem. All of them involve the minimization of a kinetic action and/or a relative entropy of some path measures with respect to the reversible Brownian motion. As the viscosity parameter $$\nu \rightarrow 0$$ we establish Gamma-convergence relations between the corresponding problems, and prove the convergence of the associated pressures arising from the incompressibility constraints. We also present new results on the time-convexity of the entropy for some of the dynamical interpolations. Along the way we extend previous results by Lavenant (Calc Var Partial Differ Equ 56(6):170, 2017) and Benamou et al. (Generalized incompressible flows, multi-marginal transport and Sinkhorn algorithm, arXiv preprint arXiv:1710.08234, 2017).

Benamou–Brenier and duality formulas for the entropic cost on $${\textsf {RCD}}^*(K,N)$$RCD∗(K,N) spaces

In this paper we prove that, within the framework of $$\textsf {RCD}^*(K,N)$$ spaces with $$N<\infty $$, the entropic cost (i.e. the minimal value of the Schrodinger problem) admits: We thus provide a complete and unifying picture of the equivalent variational representations of the Schrodinger problem as well as a perfect parallelism with the analogous formulas for the Wasserstein distance. Riemannian manifolds with Ricci curvature bounded from below are a relevant class of $$\textsf {RCD}^*(K,N)$$ spaces and our results are new even in this setting.

Ground state sign-changing solutions for elliptic equations with logarithmic nonlinearity

We study elliptic equations with logarithmic nonlinearity. With the help of the constraint variational method, quantitative deformation lemma and some new energy inequalities, we establish the existence of ground state solutions and ground state sign-changing solutions with precisely two nodal domains. Our result complements the existing ones on Schrodinger problems since the logarithmic nonlinearity is sign-changing, and satisfies neither the monotonicity condition nor Ambrosetti–Rabinowitz condition.

Theoretical and computational analysis of a nonlinear Schrödinger problem with moving boundary

In this paper, we investigate some mathematical and numerical aspects of a one-dimensional nonlinear Schrodinger problem defined in a noncylindrical domain. By a change of variable, we transform the original problem into an equivalent one defined in a cylindrical domain. To obtain the existence and uniqueness of the solution, we apply the Faedo-Galerkin method and results of compactness. The numerical simulation is performed by means of the finite element method in the associated space and the finite difference method in the temporal part, to get an approximate numerical solution. In addition, we will make an analysis of the rate of convergence of the applied methods. Finally, we will show that the results of the numerical simulation are in agreement with the theoretical analysis.

Optimal distributed control problem for cubic nonlinear Schrödinger equation

We consider an optimal internal control problem for the cubic nonlinear Schrodinger (NLS) equation on the line. We prove well-posedness of the problem and existence of an optimal control. In addition, we show first-order optimality conditions. Also, the paper includes the proof of a smoothing effect for the non-homogeneous NLS, which is necessary to obtain the existence of an optimal control.

A second order equation for Schrödinger bridges with applications to the hot gas experiment and entropic transportation cost

The Schrodinger problem is obtained by replacing the mean square distance with the relative entropy in the Monge–Kantorovich problem. It was first addressed by Schrodinger as the problem of describing the most likely evolution of a large number of Brownian particles conditioned to reach an “unexpected configuration”. Its optimal value, the entropic transportation cost, and its optimal solution, the Schrodinger bridge, stand as the natural probabilistic counterparts to the transportation cost and displacement interpolation. Moreover, they provide a natural way of lifting from the point to the measure setting the concept of Brownian bridge. In this article, we prove that the Schrodinger bridge solves a second order equation in the Riemannian structure of optimal transport. Roughly speaking, the equation says that its acceleration is the gradient of the Fisher information. Using this result, we obtain a fine quantitative description of the dynamics, and a new functional inequality for the entropic transportation cost, that generalize Talagrand’s transportation inequality. Finally, we study the convexity of the Fisher information along Schrodigner bridges, under the hypothesis that the associated reciprocal characteristic is convex. The techniques developed in this article are also well suited to study the Feynman–Kac penalisations of Brownian motion.

Global geometrical optics method for vector-valued Schrödinger problems

We extend the theory of global geometrical optics method, proposed originally for the linear scalar high-frequency wave-like equations in [Commun. Math. Sci., 2013, 11(1): 105-140], to the more general vector-valued Schrodinger problems in the semi-classical regime. The key ingredient in the global geometrical optics method is a moving frame technique in the phase space. The governing equation is transformed into a new equation but of the same type when expressed in any moving frame induced by the underlying Hamiltonian flow. The classical Wentzel-Kramers-Brillouin (WKB) analysis benefits from this treatment as it maintains valid for arbitrary but fixed evolutionary time. It turns out that a WKB-type function defined merely on the underlying Lagrangian submanifold can be obtained with the help of this moving frame technique, and from which a uniform first-order approximation of the wave field can be derived, even around caustics. The general theory is exemplified by two specific instances. One is the two-level Schrodinger system and the other is the periodic Schrodinger equation. Numerical tests validate the theoretical results.

About the analogy between optimal transport and minimal entropy

We describe some analogy between optimal transport and the Schrodinger problem where the transport cost is replaced by an entropic cost with a reference path measure. A dual Kantorovich type formulation and a Benamou-Brenier type representation formula of the entropic cost are derived, as well as contraction inequalities with respect to the entropic cost. This analogy is also illustrated with some numerical examples where the reference path measure is given by the Brownian or the Ornstein-Uhlenbeck process. Our point of view is measure theoretical and the relative entropy with respect to path measures plays a prominent role.

On elliptic string solutions in AdS3 and dS3

Classical string actions in AdS(3) and dS(3) can be connected to the sinh-Gordon and cosh-Gordon equations through Pohlmeyer reduction. We show that the problem of constructing a classical string solution with a given static or translationally invariant Pohlmeyer counterpart is equivalent to solving four pairs of effective Schrodinger problems. Each pair consists of a flat potential and an n = 1 Lame potential whose eigenvalues are connected, and, additionally, the four solutions satisfy a set of constraints. An approach for solving this system is developed by employing an interesting connection between the specific class of classical string solutions and the band structure of the Lame potential. This method is used for the construction of several families of classical string solutions, one of which turns out to be the spiky strings in AdS(3). New solutions include circular rotating strings in AdS(3) with singular time evolution of their radius and angular velocity as well as classical string solutions in dS(3).

Lazy random walks and optimal transport on graphs

This paper is about the construction of displacement interpolations on a discrete metric graph. Our approach is based on the approximation of any optimal transport problem whose cost function is a distance on a discrete graph by a sequence of Schrodinger problems associated with random walks whose jump frequencies tend down to zero. Displacement interpolations are defined as the limit of the time-marginal flows of the solutions to the Schrodinger problems. This allows to work with these interpolations by doing stochastic calculus on the approximating random walks which are regular objects, and then to pass to the limit in a slowing down procedure. The main convergence results are based on Gamma-convergence of entropy minimization problems. As a by-product, we obtain new results about optimal transport on graphs.

A stochastic nonlinear Schrödinger problem in variational formulation

This paper investigates a Schrodinger problem with power-type nonlinearity and Lipschitz-continuous diffusion term on a bounded one-dimensional domain. Using the Galerkin method and a truncation, results from stochastic partial differential equations can be applied and uniform a priori estimates for the approximations are shown. Based on these boundedness results and the structure of the nonlinearity, it follows the unique existence of the variational solution.

The estimation and profile of the critical value for a Schrödinger equation

In this paper, we are concerned with the following Schrodinger problem: , , in , where is of class . The estimation and profile of the critical value of the corresponding functional is proved, which entails the relationship between the critical value on the balls and the least-energy value on the whole space. Our results are also true for three cases of the potential function .

Discrete Geometric Approach for the Three-Dimensional Schrödinger Problem and Comparison With Finite Elements

The numerical modeling of nanoscale electron devices needs the development of accurate and efficient numerical methods, in particular, for the numerical solution of the Schrodinger problem. If FEMs allow an accurate geometric representation of the device, they lead to a discrete counterpart of Schrodinger problem in terms of a computationally heavy generalized eigenvalue problem. Exploiting the geometric structure behind the Schrodinger problem, we will construct a numerically efficient discrete counterpart of it, yielding to a standard eigenvalue problem. We will also show how the two approaches are only partially akin to each other even when lumping is applied.

A survey of the Schrödinger problem and some of its connections with optimal transport

This article is aimed at presenting the Schrodinger problem and some of its connections with optimal transport. We hope that it can be used as a basic user's guide to Schrodinger problem. We also give a survey of the related literature. In addition, some new results are proved.

Optimal Control of a Nonlinear Stochastic Schrödinger Equation

This paper deals with a nonlinear controlled Schrodinger equation which is perturbed by an additive Gaussian noise in the form of a Q-Wiener process. The nonlinearity is especially characterized by Lipschitz-continuity and growth-boundedness. Moreover, suitable initial and Neumann boundary conditions are introduced. Referring to the concept of a variational solution throughout this work, the existence and uniqueness of such a solution of the Schrodinger problem will be investigated. The main goal is to minimize a given objective functional by an optimal control. However, the objective functional depends on the solution of the controlled Schrodinger problem and the control itself. Therefore, the adjoint equation (of the stochastic Schrodinger equation) with appropriate final and Neumann boundary conditions is required. Based on a generalized solution of this adjoint problem and some important estimates concerning the difference of two solutions of the Schrodinger problem corresponding to two controls, a gradient formula in the sense of Gâteaux is calculated. Thus, a necessary optimality condition can be stated.