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.

/pdf/controlled-markov-processes-and-viscosity-solutions-4n8p6ywj5v.pdf

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.

/pdf/the-porous-medium-equation-mathematical-theory-42juib4cgk.pdf

The Porous Medium Equation: Mathematical Theory

In this paper the Dirichlet problem is considered for a wide class of quasilinear elliptic equations with C 1 coefficients in divergence form. The main result is that if Ω is an arbitrary bounded open subset of R n and ψ is a continuous function defined on the boundary of Ω, then there is a solution to the equation in Ω which assumes the boundary data continuously at all points at which a Wiener condition is satisfied. This condition is satisfied at all points of the boundary except for a set of capacity zero.

/pdf/the-wiener-criterion-and-quasilinear-uniformly-elliptic-1v2lwh3mw6.pdf

The Wiener criterion and quasilinear uniformly elliptic equations

https://www.ams.org/bull/1976-82-05/S0002-9904-1976-14098-0/S0002-9904-1976-14098-0.pdf

Review: Robert Adams, Sobolev spaces

The Dirichlet Problem for Euler–Lagrange Equations on Arbitrary Domains

The purpose of this paper is to study measure-theoretic properties of functions u belonging to a vector-valued Sobolev class on metric measure spaces that admit a Poincare inequality and are equipped with a doubling measure The properties we have selected to study are those that are closely related to those in establishing a coarea formula for u In particular, we show that both u and the graph mapping of u satisfy Luzin's condition (N) Moreover, it is shown that the graph of u is countably Hausdorff rectifiable and that the mapping u satisfies the coarea formula

The coarea formula, condition (n) and rectifiable sets for sobolev functions on metric spaces

We consider some measure-theoretic properties of functions belonging to a Sobolev-type class on metric measure spaces that admit a Poincare inequality and are equipped with a doubling measure. The properties we have selected to study are those that are related to area formulas.

William P. Ziemer

Papers

The Wiener criterion and quasilinear uniformly elliptic equations

Review: Robert Adams, Sobolev spaces

The Dirichlet Problem for Euler–Lagrange Equations on Arbitrary Domains

The coarea formula, condition (n) and rectifiable sets for sobolev functions on metric spaces

Aspects of area formulas by way of Luzin, Rad\'o, and Reichelderfer on metric measure spaces