Carmona R. Lectures on BSDEs, Stochastic Control, and Stochastic Differential Games with Financial Applications

Philadelphia: SIAM, 2016. — 363 p.This book grew out of the lecture notes I prepared for a graduate class I taught at Princeton University in 2011–12, and again in 2012–13. My goal was to introduce the students to stochastic analysis tools, which play an increasing role in the probabilistic approach to optimization problems, including stochastic control and stochastic differential games. I had invested quite a bit of effort in trying to understand the groundbreaking works of Lasry and Lions on mean field games, and of Caines, Huang, and Malham´e on Nash certainty equivalence. These initial results were intriguing, and definitely screaming for a probabilistic interpretation. While the tools of optimal control of stochastic differential systems are taught in many graduate programs in applied mathematics and operations research, I was intrigued by the fact that game theory, and especially the theory of stochastic differential games, are rarely taught in these programs. In fact, I was shocked by the lack of published literature in book form, a sad state of affairs which prompted me to write lecture notes for the class.List of Notation.
Stochastic Calculus Preliminaries.
Stochastic Differential Equations.
Notation and First Definitions.
Existence and Uniqueness of Strong Solutions: The Lipschitz Case.
SDEs of McKean–Vlasov Type.
Conditional Propagation of Chaos.
Notes & Complements.
Backward Stochastic Differential Equations.
Introduction and First Definitions.
Mean-Field BSDEs.
Reflected Backward Stochastic Differential Equations (RBSDEs).
Forward-Backward Stochastic Differential Equations (FBSDEs).
Existence and Uniqueness of Solutions.
The Affine Case.
Notes & Complements.
Stochastic Control.
Continuous Time Stochastic Optimization and Control.
Optimization of Stochastic Dynamical Systems.
First Financial Applications.
Dynamic Programming and the HJB Equation.
Infinite Horizon Case.
Constraints and Singular Control Problems.
Viscosity Solutions of HJB Equations and QVIs.
Impulse Control Problems.
Ergodic Control.
Probabilistic Approaches to Stochastic Control.
BSDEs and Stochastic Control.
Pontryagin Stochastic Maximum Principle.
Linear-Quadratic (LQ) Models.
Optimal Control of McKean–Vlasov Dynamics.
Notes & Complements.
Stochastic Differential Games.
Stochastic Differential Games.
Introduction and First Definitions.
Specific Examples.
Weak Formulation and the Case of Uncontrolled Volatility.
Game Versions of the Stochastic Maximum Principle.
A Simple Model for Systemic Risk.
A Predatory Trading Game Model.
Notes & Complements.
Mean-Field Games.
Introduction and First Definitions.
A Full Solution Without the Common Noise.
Propagation of Chaos and Approximate Nash Equilibriums.
Applications and Open Problems.
Notes & Complements.Author Index.
Subject Index.