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Optimal Stochastic Control, Stochastic Target Problems, and Backward SDE

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  • © 2013
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Overview

Authors:
  1. Nizar Touzi

    1. , Département de Mathématiques Appliquées, École Polytechnique, Palaiseau Cedex, France

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  • Provides a self-contained presentation of the recent developments in Stochastic target problems which cannot be found in any other monograph
  • Approaches quadratic backward stochastic differential equations following the point of view of Tevzadze and presented in a way to maximize the ease of understanding
  • Contains relevant examples from finance, including the Nash equilibrium example?

Part of the book series: Fields Institute Monographs (FIM, volume 29)

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Table of contents (13 chapters)

  1. Front Matter

    Pages i-x
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  2. Introduction

    • Nizar Touzi
    Pages 1-4
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  3. Conditional Expectation and Linear Parabolic PDEs

    • Nizar Touzi
    Pages 5-20
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  4. Stochastic Control and Dynamic Programming

    • Nizar Touzi
    Pages 21-37
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  5. Optimal Stopping and Dynamic Programming

    • Nizar Touzi
    Pages 39-51
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  6. Solving Control Problems by Verification

    • Nizar Touzi
    Pages 53-66
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  7. Introduction to Viscosity Solutions

    • Nizar Touzi
    Pages 67-88
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  8. Dynamic Programming Equation in the Viscosity Sense

    • Nizar Touzi
    Pages 89-99
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  9. Stochastic Target Problems

    • Nizar Touzi
    Pages 101-121
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  10. Second Order Stochastic Target Problems

    • Nizar Touzi
    Pages 123-147
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  11. Backward SDEs and Stochastic Control

    • Nizar Touzi
    Pages 149-164
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  12. Quadratic Backward SDEs

    • Nizar Touzi
    Pages 165-188
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  13. Probabilistic Numerical Methods for Nonlinear PDEs

    • Nizar Touzi
    Pages 189-199
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  14. Introduction to Finite Differences Methods

    • Agnès Tourin
    Pages 201-212
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  15. Back Matter

    Pages 213-214
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Keywords

About this book

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 secondorder 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.​

Reviews

“This is an excellent book on the topic of Stochastic Control Problems (SCP). The author transformed his notes for a graduate course at the Field Institute into a volume that will serve also as a good reference in the area. … The author has chosen the framework of diffusions, which makes the exposition more friendly and accessible to a larger audience, in particular for those who want to learn this topic.” (Jaime San Martín, Bulletin of the American Mathematical Society, Vol. 54 (2), April, 2017)

Authors and Affiliations

  • , Département de Mathématiques Appliquées, École Polytechnique, Palaiseau Cedex, France

    Nizar Touzi

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