Optimal Stopping And Free Boundary Problems
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| Author | : Goran Peskir |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 515 |
| Release | : 2006-11-10 |
| Genre | : Mathematics |
| ISBN | : 3764373903 |
This book discloses a fascinating connection between optimal stopping problems in probability and free-boundary problems. It focuses on key examples and the theory of optimal stopping is exposed at its basic principles in discrete and continuous time covering martingale and Markovian methods. Methods of solution explained range from change of time, space, and measure, to more recent ones such as local time-space calculus and nonlinear integral equations. A chapter on stochastic processes makes the material more accessible. The book will appeal to those wishing to master stochastic calculus via fundamental examples. Areas of application include financial mathematics, financial engineering, and mathematical statistics.
| Author | : Goran Peskir |
| Publisher | : Birkhäuser |
| Total Pages | : 500 |
| Release | : 2006-08-16 |
| Genre | : Business & Economics |
| ISBN | : 9783764324193 |
The book aims at disclosing a fascinating connection between optimal stopping problems in probability and free-boundary problems in analysis using minimal tools and focusing on key examples. The general theory of optimal stopping is exposed at the level of basic principles in both discrete and continuous time covering martingale and Markovian methods. Methods of solution explained range from classic ones (such as change of time, change of space, change of measure) to more recent ones (such as local time-space calculus and nonlinear integral equations). A detailed chapter on stochastic processes is included making the material more accessible to a wider cross-disciplinary audience. The book may be viewed as an ideal compendium for an interested reader who wishes to master stochastic calculus via fundamental examples. Areas of application where examples are worked out in full detail include financial mathematics, financial engineering, mathematical statistics, and stochastic analysis.
| Author | : Isabel Narra Figueiredo |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 462 |
| Release | : 2007-01-11 |
| Genre | : Mathematics |
| ISBN | : 3764377194 |
This book collects refereed lectures and communications presented at the Free Boundary Problems Conference (FBP2005). These discuss the mathematics of a broad class of models and problems involving nonlinear partial differential equations arising in physics, engineering, biology and finance. Among other topics, the talks considered free boundary problems in biomedicine, in porous media, in thermodynamic modeling, in fluid mechanics, in image processing, in financial mathematics or in computations for inter-scale problems.
| Author | : Arshak Petrosyan |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 233 |
| Release | : 2012 |
| Genre | : Mathematics |
| ISBN | : 0821887947 |
The regularity theory of free boundaries flourished during the late 1970s and early 1980s and had a major impact in several areas of mathematics, mathematical physics, and industrial mathematics, as well as in applications. Since then the theory continued to evolve. Numerous new ideas, techniques, and methods have been developed, and challenging new problems in applications have arisen. The main intention of the authors of this book is to give a coherent introduction to the study of the regularity properties of free boundaries for a particular type of problems, known as obstacle-type problems. The emphasis is on the methods developed in the past two decades. The topics include optimal regularity, nondegeneracy, rescalings and blowups, classification of global solutions, several types of monotonicity formulas, Lipschitz, $C^1$, as well as higher regularity of the free boundary, structure of the singular set, touch of the free and fixed boundaries, and more. The book is based on lecture notes for the courses and mini-courses given by the authors at various locations and should be accessible to advanced graduate students and researchers in analysis and partial differential equations.
| Author | : Nizar Touzi |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 219 |
| Release | : 2012-09-25 |
| Genre | : Mathematics |
| ISBN | : 1461442869 |
This book collects some recent developments in stochastic control theory with applications to financial mathematics. We first address standard stochastic control problems from the viewpoint of the recently developed weak dynamic programming principle. A special emphasis is put on the regularity issues and, in particular, on the behavior of the value function near the boundary. We then provide a quick review of the main tools from viscosity solutions which allow to overcome all regularity problems. We next address the class of stochastic target problems which extends in a nontrivial way the standard stochastic control problems. Here the theory of viscosity solutions plays a crucial role in the derivation of the dynamic programming equation as the infinitesimal counterpart of the corresponding geometric dynamic programming equation. The various developments of this theory have been stimulated by applications in finance and by relevant connections with geometric flows. Namely, the second order extension was motivated by illiquidity modeling, and the controlled loss version was introduced following the problem of quantile hedging. The third part specializes to an overview of Backward stochastic differential equations, and their extensions to the quadratic case.
| Author | : Kumar Muthuraman |
| Publisher | : Now Publishers Inc |
| Total Pages | : 94 |
| Release | : 2008 |
| Genre | : Boundary value problems |
| ISBN | : 1601981686 |
Outlines and explains a recent computational method that solves free boundary problems by reducing them into a sequence of fixed boundary problems which are relatively easy to solve numerically.
| Author | : Yuan Shih Chow |
| Publisher | : Dover Publications |
| Total Pages | : 139 |
| Release | : 1991-01 |
| Genre | : Mathematics |
| ISBN | : 9780486666501 |
| Author | : Gioia Carinci |
| Publisher | : Springer |
| Total Pages | : 106 |
| Release | : 2016-06-22 |
| Genre | : Mathematics |
| ISBN | : 3319333704 |
In this volume a theory for models of transport in the presence of a free boundary is developed.Macroscopic laws of transport are described by PDE's. When the system is open, there are several mechanisms to couple the system with the external forces. Here a class of systems where the interaction with the exterior takes place in correspondence of a free boundary is considered. Both continuous and discrete models sharing the same structure are analysed. In Part I a free boundary problem related to the Stefan Problem is worked out in all details. For this model a new notion of relaxed solution is proposed for which global existence and uniqueness is proven. It is also shown that this is the hydrodynamic limit of the empirical mass density of the associated particle system. In Part II several other models are discussed. The expectation is that the results proved for the basic model extend to these other cases.All the models discussed in this volume have an interest in problems arising in several research fields such as heat conduction, queuing theory, propagation of fire, interface dynamics, population dynamics, evolution of biological systems with selection mechanisms.In general researchers interested in the relations between PDE’s and stochastic processes can find in this volume an extension of this correspondence to modern mathematical physics.
| Author | : Stanislav Nikolaevich Antont︠s︡ev |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 372 |
| Release | : 1992 |
| Genre | : Mathematics |
| ISBN | : 9783764327842 |
Some extremum and unilateral boundary value problems in viscous hydrodynamics.- On axisymmetric motion of the fluid with a free surface.- On the occurrence of singularities in axisymmetrical problems of hele-shaw type.- New asymptotic method for solving of mixed boundary value problems.- Some results on the thermistor problem.- New applications of energy methods to parabolic and elliptic free boundary problems.- A localized finite element method for nonlinear water wave problems.- Approximate method of investigation of normal oscillations of viscous incompressible liquid in container.- The classical Stefan problem as the limit case of the Stefan problem with a kinetic condition at the free boundary.- A mathematical model of oscillations energy dissipation of viscous liquid in a tank.- Existence of the classical solution of a two-phase multidimensional Stefan problem on any finite time interval.- Asymptotic theory of propagation of nonstationary surface and internal waves over uneven bottom.- Multiparametric problems of two-dimensional free boundary seepage.- Nonisothermal two-phase filtration in porous media.- Explicit solution of time-dependent free boundary problems.- Nonequilibrium phase transitions in frozen grounds.- System of variational inequalities arising in nonlinear diffusion with phase change.- Contact viscoelastoplastic problem for a beam.- Application of a finite-element method to two-dimensional contact problems.- Computations of a gas bubble motion in liquid.- Waves on the liquid-gas free surface in the presence of the acoustic field in gas.- Smooth bore in a two-layer fluid.- Numerical calculation of movable free and contact boundaries in problems of dynamic deformation of viscoelastic bodies.- On the canonical variables for two-dimensional vortex hydrodynamics of incompressible fluid.- About the method with regularization for solving the contact problem in elasticity.- Space evolution of tornado-like vortex core.- Optimal shape design for parabolic system and two-phase Stefan problem.- Incompressible fluid flows with free boundary and the methods for their research.- On the Stefan problems for the system of equations arising in the modelling of liquid-phase epitaxy processes.- Stefan problem with surface tension as a limit of the phase field model.- The modelization of transformation phase via the resolution of an inclusion problem with moving boundary.- To the problem of constructing weak solutions in dynamic elastoplasticity.- The justification of the conjugate conditions for the Euler's and Darcy's equations.- On an evolution problem of thermo-capillary convection.- Front tracking methods for one-dimensional moving boundary problems.- On Cauchy problem for long wave equations.- On fixed point (trial) methods for free boundary problems.- Nonlinear theory of dynamics of a viscous fluid with a free boundary in the process of a solid body wetting.
| Author | : Luis Angel Caffarelli |
| Publisher | : Edizioni della Normale |
| Total Pages | : 0 |
| Release | : 1999-10-01 |
| Genre | : Mathematics |
| ISBN | : 9788876422492 |
The material presented here corresponds to Fermi lectures that I was invited to deliver at the Scuola Normale di Pisa in the spring of 1998. The obstacle problem consists in studying the properties of minimizers of the Dirichlet integral in a domain D of Rn, among all those configurations u with prescribed boundary values and costrained to remain in D above a prescribed obstacle F. In the Hilbert space H1(D) of all those functions with square integrable gradient, we consider the closed convex set K of functions u with fixed boundary value and which are greater than F in D. There is a unique point in K minimizing the Dirichlet integral. That is called the solution to the obstacle problem.
