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 study here a classical image denoising technique introduced by L. Rudin and S. Osher a few years ago, namely the constrained minimization of the total variation (TV) of the image. First, we give results of existence and uniqueness and prove the link between the constrained minimization problem and the minimization of an associated Lagrangian functional. Then we describe a relaxation method for computing the solution, and give a proof of convergence. After this, we explain why the TV-based model is well suited to the recovery of some images and not of others. We eventually propose an alternative approach whose purpose is to handle the minimization of the minimum of several convex functionals. We propose for instance a variant of the original TV minimization problem that handles correctly some situations where TV fails.

Image recovery via total variation minimization and related problems

We propose simple and extremely efficient methods for solving the basis pursuit problem $\min\{\|u\|_1 : Au = f, u\in\mathbb{R}^n\},$ which is used in compressed sensing. Our methods are based on Bregman iterative regularization, and they give a very accurate solution after solving only a very small number of instances of the unconstrained problem $\min_{u\in\mathbb{R}^n} \mu\|u\|_1+\frac{1}{2}\|Au-f^k\|_2^2$ for given matrix $A$ and vector $f^k$. We show analytically that this iterative approach yields exact solutions in a finite number of steps and present numerical results that demonstrate that as few as two to six iterations are sufficient in most cases. Our approach is especially useful for many compressed sensing applications where matrix-vector operations involving $A$ and $A^\top$ can be computed by fast transforms. Utilizing a fast fixed-point continuation solver that is based solely on such operations for solving the above unconstrained subproblem, we were able to quickly solve huge instances of compressed sensing problems on a standard PC.

https://epubs.siam.org/doi/pdf/10.1137/070703983

Bregman Iterative Algorithms for $\ell_1$-Minimization with Applications to Compressed Sensing

Geodesics in subriemannian manifolds: Dido meets Heisenberg Chow's theorem: Getting from A to B A remarkable horizontal curve Curvature and nilpotentization Singular curves and geodesics A zoo of distributions Cartan's approach The tangent cone and Carnot groups Discrete groups tending to Carnot geometries Open problems Mechanics and geometry of bundles: Metrics on bundles Classical particles in Yang-Mills fields Quantum phases Falling, swimming, and orbiting Appendices: Geometric mechanics Bundles and the Hopf fibration The Sussmann and Ambrose-Singer theorems Calculus of the endpoint map and existence of geodesics Bibliography Index.

A Tour of Subriemannian Geometries, Their Geodesics and Applications

The Heat Equation is one of the three classical linear partial differential equations of second order that form the basis of any elementary introduction to the area of PDEs, and only recently has it come to be fairly well understood. In this monograph, aimed at research students and academics in mathematics and engineering, as well as engineering specialists, Professor Vazquez provides a systematic and comprehensive presentation of the mathematical theory of the nonlinear heat equation usually called the Porous Medium Equation (PME). This equation appears in a number of physical applications, such as to describe processes involving fluid flow, heat transfer or diffusion. Other applications have been proposed in mathematical biology, lubrication, boundary layer theory, and other fields. Each chapter contains a detailed introduction and is supplied with a section of notes, providing comments, historical notes or recommended reading, and exercises for the reader.

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The Porous Medium Equation: Mathematical Theory

Weakly differentiable functions

Preliminaries Potential theory Quasilinear equations Fine regularity theory Variational inequalities--Regularity Existence theory References Index Notation index.

Fine Regularity of Solutions of Elliptic Partial Differential Equations

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Extremal length and conformal capacity

In this paper we are concerned with the behavior at the boundary of weak solutions of the Dirichlet problem for quasilinear elliptic equations of second order in an open set O ~ IR". The main result is that a bounded weak solution is continuous at a boundary point of f2 provided that the complement of f2 in a neighborhood of this point is sufficiently " thick" when measured by an appropriate capacity. In the case of Laplace's equation this condition reduces to that considered by WIENER [W1], [W2"]. The equations considered are of the general divergence structure type

A regularity condition at the boundary for solutions of quasilinear elliptic equations

This paper is motivated by the observation that solutions to certain variational inequalities involving partial differential operators of the form divA(x, u, Vu) + B(x, u, Vu), where A and B are Borel measurable, are solutions to the equation divA(x, u, Vu) + B(x, u, Vu) = ,u for some nonnegative Radon measure ,i. Among other things, it is shown that if u is a Holder continuous solution to this equation, then the measure ,u satisfies the growth property j[B(x, r)] 1 is given by the structure of the differential operator. Conversely, if u is assumed to satisfy this growth condition, then it is shown that u satisfies a Harnack-type inequality, thus proving that u is locally bounded. Under -the additional assumption that A is strongly monotonic, it is shown that u is Holder continuous.

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William P. Ziemer

Papers

Weakly differentiable functions

Fine Regularity of Solutions of Elliptic Partial Differential Equations

Extremal length and conformal capacity

A regularity condition at the boundary for solutions of quasilinear elliptic equations

Local behavior of solutions of quasilinear elliptic equations with general structure