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Convex Analysisの二,三の進展について

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.

Stochastic Equations in Infinite Dimensions

This book is intended as an introduction to optimal stochastic control for continuous time Markov processes and to the theory of viscosity solutions. The authors approach stochastic control problems by the method of dynamic programming. The text provides an introduction to dynamic programming for deterministic optimal control problems, as well as to the corresponding theory of viscosity solutions. A new Chapter X gives an introduction to the role of stochastic optimal control in portfolio optimization and in pricing derivatives in incomplete markets. Chapter VI of the First Edition has been completely rewritten, to emphasize the relationships between logarithmic transformations and risk sensitivity. A new Chapter XI gives a concise introduction to two-controller, zero-sum differential games. Also covered are controlled Markov diffusions and viscosity solutions of Hamilton-Jacobi-Bellman equations. The authors have tried, through illustrative examples and selective material, to connect stochastic control theory with other mathematical areas (e.g. large deviations theory) and with applications to engineering, physics, management, and finance. In this Second Edition, new material on applications to mathematical finance has been added. Concise introductions to risk-sensitive control theory, nonlinear H-infinity control and differential games are also included.

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Controlled Markov processes and viscosity solutions

We show that various inverse problems in signal recovery can be formulated as the generic problem of minimizing the sum of two convex functions with certain regularity properties. This formulation makes it possible to derive existence, uniqueness, characterization, and stability results in a unified and standardized fashion for a large class of apparently disparate problems. Recent results on monotone operator splitting methods are applied to establish the convergence of a forward-backward algorithm to solve the generic problem. In turn, we recover, extend, and provide a simplified analysis for a variety of existing iterative methods. Applications to geometry/texture image decomposition schemes are also discussed. A novelty of our framework is to use extensively the notion of a proximity operator, which was introduced by Moreau in the 1960s.

/pdf/signal-recovery-by-proximal-forward-backward-splitting-41tfo8pc7o.pdf

Signal recovery by proximal forward-backward splitting ∗

This monograph presents a summary account of the subject of a posteriori error estimation for finite element approximations of problems in mechanics. The study primarily focuses on methods for linear elliptic boundary value problems. However, error estimation for unsymmetrical systems, nonlinear problems, including the Navier-Stokes equations, and indefinite problems, such as represented by the Stokes problem are included. The main thrust is to obtain error estimators for the error measured in the energy norm, but techniques for other norms are also discussed.

A Posteriori Error Estimation in Finite Element Analysis

Applied Nonlinear Analysis

Viability theory

0 Background Notes- 1 Continuous Partitions of Unity- 2 Absolutely Continuous Functions- 3 Some Compactness Theorems- 4 Weak Convergence and Asymptotic Center of Bounded Sequences- 5 Closed Convex Hulls and the Mean-Value Theorem- 6 Lower Semicontinuous Convex Functions and Projections of Best Approximation- 7 A Concise Introduction to Convex Analysis- 1 Set-Valued Maps- 1 Set-Valued Maps and Continuity Concepts- 2 Examples of Set-Valued Maps- 3 Continuity Properties of Maps with Closed Convex Graph- 4 Upper Hemicontinuous Maps and the Convergence Theorem- 5 Hausdorff Topology- 6 The Selection Problem- 7 The Minimal Selection- 8 Chebishev Selection- 9 The Barycentric Selection- 10 Selection Theorems for Locally Selectionable Maps- 11 Michael's Selection Theorem- 12 The Approximate Selection Theorem and Kakutani's Fixed Point Theorem- 13 (7-Selectionable Maps- 14 Measurable Selections- 2 Existence of Solutions to Differential Inclusions- 1 Convex Valued Differential Inclusions- 2 Qualitative Properties of the Set of Trajectories of Convex-Valued Differential Inclusions- 3 Nonconvex-Valued Differential Inclusions- 4 Differential Inclusions with Lipschitzean Maps and the Relaxation Theorem- 5 The Fixed-Point Approach- 6 The Lower Semicontinuous Case- 3 Differential Inclusions with Maximal Monotone Maps- 1 Maximal Monotone Maps- 2 Existence and Uniqueness of Solutions to Differential Inclusions with Maximal Monotone Maps- 3 Asymptotic Behavior of Trajectories and the Ergodic Theorem- 4 Gradient Inclusions- 5 Application: Gradient Methods for Constrained Minimization Problems- 4 Viability Theory: The Nonconvex Case- 1 Bouligand's Contingent Cone- 2 Viable and Monotone Trajectories- 3 Contingent Derivative of a Set-Valued Map- 4 The Time Dependent Case- 5 A Continuous Version of Newton's Method- 6 A Viability Theorem for Continuous Maps with Nonconvex Images- 7 Differential Inclusions with Memory- 5 Viability Theory and Regulation of Controled Systems: The Convex Case- 1 Tangent Cones and Normal Cones to Convex Sets- 2 Viability Implies the Existence of an Equilibrium- 3 Viability Implies the Existence of Periodic Trajectories- 4 Regulation of Controled Systems Through Viability- 5 Walras Equilibria and Dynamical Price Decentralization- 6 Differential Variational Inequalities- 7 Rate Equations and Inclusions- 6 Liapunov Functions- 1 Upper Contingent Derivative of a Real-Valued Function- 2 Liapunov Functions and Existence of Equilibria- 3 Monotone Trajectories of a Differential Inclusion- 4 Construction of Liapunov Functions- 5 Stability and Asymptotic Behavior of Trajectories- Comments

Differential Inclusions: Set-Valued Maps and Viability Theory

This book presents a unified treatment of optimization theory, game theory and a general equilibrium theory in economics in the framework of nonlinear functional analysis. It not only provides powerful and versatile tools for solving specific problems in economics and the social sciences but also serves as a unifying theme in the mathematical theory of these subjects as well as in pure mathematics itself.

Mathematical methods of game and economic theory

This chapter relates the notions of mutations with the concept of graphical derivatives of set-valued maps and more generally links the above results of morphological analysis with some basic facts of set-valued analysis that we shall recall.

Jean-Pierre Aubin

Papers

Applied Nonlinear Analysis

Viability theory

Differential Inclusions: Set-Valued Maps and Viability Theory

Mathematical methods of game and economic theory

Set-valued analysis