René Carmona

Bio: René Carmona is an academic researcher from Princeton University. The author has contributed to research in topics: Stochastic differential equation & Nash equilibrium. The author has an hindex of 53, co-authored 206 publications receiving 10163 citations. Previous affiliations of René Carmona include University of California, Irvine.

Spectral Theory of Random Schrödinger Operators

Abstract: I Spectral Theory of Self-Adjoint Operators.- 1 Domains, Adjoints, Resolvents and Spectra.- 2 Resolutions of the Identity.- 3 Representation Theorems.- 4 The Spectral Theorem.- 5 Quadratic Forms and Self-adjoint Operators.- 6 Self-adjoint Extensions of Symmetric Operators.- 7 Problems.- 8 Notes and Complements.- II Schrodinger Operators.- 1 The Free Hamiltonians.- 2 Schrodinger Operators as Perturbations.- 2.1 Self-adjointness.- 2.2 Perturbation of the Absolutely Continuous Spectrum.- 2.3 An Approximation Argument.- 3 Path Integral Formulas.- 3.1 Brownian Motions and the Free Hamiltonians.- 3.2 The Feynman-Kac Formula.- 4 Eigenfunctions.- 4.1 L2-Eigenfunctions.- 4.2 The Periodic Case.- 4.3 Generalized Eigenfunction Expansions.- 5 Problems.- 6 Notes and Complements.- III One-Dimensional Schrodinger Operators.- 1 The Continuous Case.- 1.1 Essential Self-adjointness.- 1.2 The Operator in an Interval.- 1.3 Green's and Weyl-Titchmarsh's Functions.- 1.4 The Propagator.- 1.5 Examples.- 2 The Lattice Case.- 3 Approximations of the Spectral Measures.- 4 Spectral Types.- 4.1 Absolutely Continuous Spectrum.- 4.2 Singular Spectrum.- 4.3 Pure Point Spectrum.- 5 Quasi-one Dimensional Schrodinger Operators.- 5.1 The Schrodinger Operator in a Strip.- 5.2 Approximation of the Spectral Measures.- 5.3 Nature of the Spectrum.- 6 Problems.- 7 Notes and Complements.- IV Products of Random Matrices.- 1 General Ergodic Theorems.- 2 Matrix Valued Systems.- 3 Group Action on Compact Spaces.- 3.1 Definitions and Notations.- 3.2 Laplace Operators on the Space of Continuous Functions.- 3.3 The Laplace Operators on the Space of Holder Continuous Functions.- 4 Products of Independent Random Matrices.- 4.1 The Upper Lyapunov Exponent.- 4.2 The Lyapunov Spectrum.- 4.3 Schrodinger Matrices.- 5 Markovian Multiplicative Systems.- 5.1 The Upper Lyapunov Exponent.- 5.2 The Lyapunov Spectrum.- 5.3 Laplace Transform.- 6 Boundaries of the Symplectic Group.- 7 Problems.- 8 Notes and Comments.- V Ergodic Families of Self-Adjoint Operators.- 1 Measurability Concepts.- 2 Spectra of Ergodic Families.- 3 The Case of Random Schrodinger Operators.- 3.1 Examples.- 4 Regularity Properties of the Lyapunov Exponents.- 4.1 Subharmonicity.- 4.2 Continuity.- 4.3 Local Holder Continuity.- 4.4 Smoothness.- 5 Problems.- 6 Notes and Complements.- VI The Integrated Density of States.- 1 Existence Problems.- 1.1 Setting of the Problem.- 1.2 Path Integral Approach.- 1.3 Functional Analytic Approach.- 2 Asymptotic Behavior and Lifschitz Tails.- 2.1 Tauberian Arguments.- 2.2 The Anderson Model.- 3 More on the Lattice Case.- 4 The One Dimensional Cases.- 4.1 The Continuous Case.- 4.2 The Lattice Case.- 5 Problems.- 6 Notes and Complements.- VII Absolutely Continuous Spectrum and Inverse Theory.- 1 The w-function.- 1.1 More on Herglotz Functions.- 1.2 The Continuous Case.- 1.3 The Lattice Case.- 2 Periodic and Almost Periodic Potentials.- 2.1 Floquet Theory.- 2.2 Inverse Spectral Theory.- 2.3 The Lattice Case.- 2.4 Almost Periodic Potentials.- 3 The Absolutely Continuous Spectrum.- 3.1 The Essential Support of the Absolutely Continuous Spectrum.- 3.2 Support Theorems and Deterministic Potentials.- 4 Inverse Spectral Theory.- 4.1 The Continuous Case.- 4.2 The Lattice Case.- 5 Miscellaneous.- 5.1 Potentials Taking Finitely Many Values.- 5.2 A Remark on the Multidimensional Case.- 6 Problems.- 7 Notes and Complements.- VIII Localization in One Dimension.- 1 Pointwise Theory.- 1.1 Kotani's Trick.- 1.2 The Discrete Case.- 1.3 The General Case.- 2 Perturbation Theory.- 3 Operator Theory.- 3.1 The Discrete I.I.D. Model.- 3.2 The Markov Model.- 3.3 The Discrete I.I.D. Model on the Strip.- 4 Localization for Singular Potentials.- 5 Non-Stationary Processes.- 5.1 The Discrete Case.- 5.2 The Continuous Case.- 6 Problems.- 7 Notes and Complements.- IX Localization in Any Dimension.- 1 Exponential Decay of the Green's Function at Fixed Energy.- 1.1 Decay of the Green's Function in Boxes.- 1.2 Decay of the Green's Function in ?d.- 2 Localization for A.C. Potentials.- 2.1 Pointwise Theory.- 2.2 Perturbation Theory.- 3 A Direct Proof of Localization.- 3.1 Examples.- 3.2 The Proof.- 3.3 Extensions.- 4 Problems.- 5 Notes and Complements.- Notation Index.

Probabilistic theory of mean field games with applications

Abstract: This two-volume book offers a comprehensive treatment of the probabilistic approach to mean field game models and their applications. The book is self-contained in nature and includes original material and applications with explicit examples throughout, including numerical solutions. Volume II tackles the analysis of mean field games in which the players are affected by a common source of noise. The first part of the volume introduces and studies the concepts of weak and strong equilibria, and establishes general solvability results. The second part is devoted to the study of the master equation, a partial differential equation satisfied by the value function of the game over the space of probability measures. Existence of viscosity and classical solutions are proven and used to study asymptotics of games with finitely many players. Together, both Volume I and Volume II will greatly benefit mathematical graduate students and researchers interested in mean field games. The authors provide a detailed road map through the book allowing different access points for different readers and building up the level of technical detail. The accessible approach and overview will allow interested researchers in the applied sciences to obtain a clear overview of the state of the art in mean field games.

Anderson localization for Bernoulli and other singular potentials

Abstract: We prove exponential localization in the Anderson model under very weak assumptions on the potential distribution. In one dimension we allow any measure which is not concentrated on a single point and possesses some finite moment. In particular this solves the longstanding problem of localization for Bernoulli potentials (i.e., potentials that take only two values). In dimensions greater than one we prove localization at high disorder for potentials with Holder continuous distributions and for bounded potentials whose distribution is a convex combination of a Holder continuous distribution with high disorder and an arbitrary distribution. These include potentials with singular distributions. We also show that for certain Bernoulli potentials in one dimension the integrated density of states has a nontrivial singular component.

Pricing and Hedging Spread Options

Abstract: We survey theoretical and computational problems associated with the pricing and hedging of spread options. These options are ubiquitous in the financial markets, whether they be equity, fixed income, foreign exchange, commodities, or energy markets. As a matter of introduction, we present a general overview of the common features of all spread options by discussing in detail their roles as speculation devices and risk management tools. We describe the mathematical framework used to model them, and we review the numerical algorithms actually used to price and hedge them. There is already extensive literature on the pricing of spread options in the equity and fixed income markets, and our contribution is mostly to put together material scattered across a wide spectrum of recent textbooks and journal articles. On the other hand, information about the various numerical procedures that can be used to price and hedge spread options on physical commodities is more difficult to find. For this reason, we make a systematic effort to choose examples from the energy markets in order to illustrate the numerical challenges associated with these instruments. This gives us a chance to discuss an interesting application of spread options to an asset valuation problem after it is recast in the framework of real options. This approach is currently the object of intense mathematical research. In this spirit, we review the two major avenues to modeling energy price dynamics. We explain how the pricing and hedging algorithms can be implemented in the framework of models for both the spot price dynamics and the forward curve dynamics.

A wavelet tour of signal processing

Abstract: Introduction to a Transient World. Fourier Kingdom. Discrete Revolution. Time Meets Frequency. Frames. Wavelet Zoom. Wavelet Bases. Wavelet Packet and Local Cosine Bases. An Approximation Tour. Estimations are Approximations. Transform Coding. Appendix A: Mathematical Complements. Appendix B: Software Toolboxes.

Stochastic Equations in Infinite Dimensions

Abstract: Part I. Foundations: 1. Random variables 2. Probability measures 3. Stochastic processes 4. The stochastic integral Part II. Existence and Uniqueness: 5. Linear equations with additive noise 6. Linear equations with multiplicative noise 7. Existence and uniqueness for nonlinear equations 8. Martingale solutions Part III. Properties of Solutions: 9. Markov properties and Kolmogorov equations 10. Absolute continuity and Girsanov's theorem 11. Large time behaviour of solutions 12. Small noise asymptotic.

Wavelet Methods for Time Series Analysis

Abstract: 1. Introduction to wavelets 2. Review of Fourier theory and filters 3. Orthonormal transforms of time series 4. The discrete wavelet transform 5. The maximal overlap discrete wavelet transform 6. The discrete wavelet packet transform 7. Random variables and stochastic processes 8. The wavelet variance 9. Analysis and synthesis of long memory processes 10. Wavelet-based signal estimation 11. Wavelet analysis of finite energy signals Appendix. Answers to embedded exercises References Author index Subject index.

Stochastic Differential Equations

Abstract: We explore in this chapter questions of existence and uniqueness for solutions to stochastic differential equations and offer a study of their properties. This endeavor is really a study of diffusion processes. Loosely speaking, the term diffusion is attributed to a Markov process which has continuous sample paths and can be characterized in terms of its infinitesimal generator.