Optimal Stochastic Control, Stochastic Target Problems, and Backward SDE

1. Front Matter

   Pages i-x

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2. Introduction

   • Nizar Touzi

   Pages 1-4

3. Conditional Expectation and Linear Parabolic PDEs

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   Pages 5-20

4. Stochastic Control and Dynamic Programming

   • Nizar Touzi

   Pages 21-37

5. Optimal Stopping and Dynamic Programming

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   Pages 39-51

6. Solving Control Problems by Verification

   • Nizar Touzi

   Pages 53-66

7. Introduction to Viscosity Solutions

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   Pages 67-88

8. Dynamic Programming Equation in the Viscosity Sense

   • Nizar Touzi

   Pages 89-99

9. Stochastic Target Problems

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   Pages 101-121

10. Second Order Stochastic Target Problems

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    Pages 123-147

11. Backward SDEs and Stochastic Control

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    Pages 149-164

12. Quadratic Backward SDEs

    • Nizar Touzi

    Pages 165-188

13. Probabilistic Numerical Methods for Nonlinear PDEs

    • Nizar Touzi

    Pages 189-199

14. Introduction to Finite Differences Methods

    • Agnès Tourin

    Pages 201-212

15. Back Matter

    Pages 213-214

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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.​

• backwards stochastic differential equations
• dynamic programming
• financial mathematics
• stochastic control problems
• stochastic control theory
• viscosity solutions
• quantitative finance
• partial differential equations

“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)

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

  Nizar Touzi

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