New Models and Methods in Dynamic Portfolio Optimization

This book presents some new models and methods in the context of dynamical portfolio optimization. It encapsulates the authors' recent progress in their research on several interesting, featured issues of dynamic portfolio optimization problems with default contagion, tracking benchmark, consumption habit, and model uncertainty.

• These models include the default the contagion model with infinite regime-switching under complete information and partial information; portfolio optimization model with consumption habit formation; optimal tracking model; extended Merton's problem with relaxed benchmark tracking and reinforcement learning in optimal tracking portfolio.
• The methods for addressing these problems are by developing the monotone dynamical system, martingale representation theorem under partial information, quadratic BSDE with jumps, convex duality method, decomposition-homogenization technique of Neumann problem, stochastic flow, and q-function learning with state reflection.

For the sake of the reader's convenience, preliminary knowledge on stochastic analysis and stochastic control are summarized in Chapters 2 and 3, which also serve as a brief basic introduction to the theory of SDEs, BSDEs, and the theory of optimal stochastic control.

The book will be a good reference for graduate students and researchers working on stochastic control and mathematical finance. The reader may pursue some presented research problems and be inspired to formulate and study other new and interesting problems in dynamic portfolio optimization and beyond.

• Preface
• Notation
• Acknowledgments
• Introduction:
  • Simple Examples
  • Perspective
  • Overview
• Preliminary:
  • Stochastic Differential Equations
  • Hybrid Diffusion System
  • Feymann–Kac's Formula
  • Backward Stochastic Differential Equations
• Elements of Stochastic Control:
  • Formulation of Stochastic Control Problem
  • Dynamic Programming Principle
  • Hamilton–Jacobi–Bellman Equations
  • Stochastic Maximum Principle
  • Risk Sensitive Stochastic Control
• Risk-Sensitive Credit Portfolio Optimization: Complete Information:
  • The Market Model
  • Formulation of Portfolio Optimization Problem
  • Recursive HJB Equations
  • Well-Posedness of Recursive HJB Equations
  • The Verification Result
  • The Convergence Rate
• Risk-Sensitive Credit Portfolio Optimization: Partial Information:
  • The Market Model
  • Partial Information and Filter Processes
  • Martingale Representation Theorem
  • General Correlation Volatility Matrix
  • Risk-Sensitive Control under Partial Information
  • Quadratic BSDE with Jumps
  • The Optimal Investment Strategy
  • Uniqueness of Solutions to BSDE
• Portfolio Optimization with Consumption Habit in Incomplete Market:
  • The Market Model
  • The Characterization of Financeable Consumption
  • The Dual Optimization Problem and Its Solvability
  • Proofs of Main Results
• Optimal Tracking Problem in Portfolio Optimization:
  • The Market Model
  • Formulation of Optimal Tracking Problem
  • Auxiliary Stochastic Control Problem
  • The Linearization of HJB Equation
  • Probabilistic Solution of Linear Dual HJB Equation
  • Verification Results
  • The GBM Benchmark Process
• Extended Merton's Problem with Relaxed Benchmark Tracking:
  • The Market Model
  • Equivalent Stochastic Control Problem
  • Solvability of the Dual PDE
  • Verification Theorem
• Reinforcement Learning for Optimal Tracking Portfolio in Incomplete Markets:
  • The Market Model
  • The Equivalent Control Problem
  • The Continuous Time Reinforcement Learning (RL)
  • Numerical Examples
  • Proofs of Main Results