Chapter 1．Basic Stochastic Calculus

1．Probability

1.1．Probability spaces

1.2．Random variables

1.3．Conditional expectation

1.4．Convergence of probabilities

2．Stochastic Processes

2.1．General considerations

2.2．Brownian motions

3．Stopping Times

4．Martingales

5．It？'s Integral

5.1．Nondifferentiability of Brownian motion

5.2．Definition of It？'s integral and basic properties

5.3．It？'s formula

5.4．Martingale representation theorems

6．Stochastic Differential Equations

6.1．Strong solutions

6.2．Weak solutions

6.3．Linear SDEs

6.4．Other types of SDEs

Chapter 2．Stochastic Optimal Control Problems

1．Introduction

2．Deterministic Cases Revisited

3．Examples of Stochastic Control Problems

3.1．Production planning

3.2．Investment vs.consumption

3.3．Reinsurance and dividend management

3.4．Technology diffusion

3.5．Queueing systems in heavy traffic

4．Formulations of Stochastic Optimal Control Problems

4.1．Strong formulation

4.2．Weak formulation

5．Existence of Optimal Controls

5.1．A deterministic result

5.2．Existence under strong formulation

5.3．Existence under weak formulation

6．Reachable Sets of Stochastic Control Systems

6.1．Nonconvexity of the reachable sets

6.2．Noncloseness of the reachable sets

7．Other Stochastic Control Models

7.1．Random duration

7.2．Optimal stopping

7.3．Singular and impulse controls

7.4．Risk-sensitive controls

7.5．Ergodic controls

7.6．Partially observable systems

8．Historical Remarks

Chapter 3．Maximum Principle and Stochastic Hamiltonian Systems

1．Introduction

2．The Deterministic Case Revisited

3．Statement of the Stochastic Maximum Principle

3.1．Adjoint equations

3.2．The maximum principle and stochastic Hamiltonian systems

3.3．A worked-out example

4．A Proof of the Maximum Principle

4.1．A moment estimate

4.2．Taylor expansions

4.3．Duality analysis and completion of the proof

5．Sufficient Conditions of Optimality

6．Problems with Statc Constraints

6.1．Formulation of the problem and the maximum principle

6.2．Some preliminary lemmas

6.3．A proof of Theorem 6.1

7．Historical Remarks

Chapter 4．Dynamic Programming and HJB Equations

1．Introduction

2．The Detcrministic Case Revisited

3．The Stochastic Principle of Optimality and the HJB Equation

3.1．A stochastic framework for dynamic programming

3.2．Principle of optimality

3.3．The HJB equation

4．Other Properties of the Value Function

4.1．Continuous dependence on parameters

4.2．Semiconcavity

5．Viscosity Solutions

5.1．Definitions

5.2．Some properties

6．Uniqueness of Viscosity Solutions

6.1．A uniqueness theorem

6.2．Proofs of Lemmas 6.6 and 6.7

7．Historical Remarks

Chapter 5．The Relationship Between the Maximum Principle and Dynamic Programming

1．Introduction

2．Classical Hamilton-Jacobi Thcory

3．Relationship for Deterministic Systems

3.1．Adjoint variable and value function:Smooth case

3.2．Economic interpretation

3.3．Methods of characteristics and the Feynman-Kac formula

3.4．Adjoint variable and value function:Nonsmooth case

3.5．Vcrification theorems

4．Relationship for Stochastic Systems

4.1．Smooth case

4.2．Nonsmooth case:Differentials in the spatial variable

4.3．Nonsmooth case:Differentials in the time variable

5．Stochastic Vcrification Theorems

5.1．Smooth case

5.2．Nonsmooth case

6．Optimal Feedback Controls

7．Historical Remarks

Chapter 6．Linear Quadratic Optimal Control Problems

1．Introduction

2．The Deterministic LQ Problems Revisited

2.1．Formulation

2.2．A minimization problem of a quadratic functional

2.3．A linear Hamiltonian system

2.4．The Riccati equation and feedback optimal control

3．Formulation of Stochastic LQ Problems

3.1．Statement of the problems

3.2．Examples

4．Finiteness and Solvability

5．A Necessary Condition and a Hamiltonian System

6．Stochastic Riccati Equations

7．Global Solvability of Stochastic Riccati Equations

7.1．Existence:The standard case

7.2．Existence:The case C=0，S=0，and Q,G≥0

7.3．Existence:The one-dimensional case

8．A Mean-variance Portfolio Selection Problem

9．Historical Rcmarks

Chapter 7．Backward Stochastic Differential Equations

1．Introduction

2．Linear Backward Stochastic Diffrential Equations

3．Nonlinear Backward Stochastic Differential Equations

3.1．BSDEs in finite deterministic durations:Method of contraction mapping

3.2．BSDEs in random durations:Method of continuation

4．Feynman-Kac-Type Formulae

4.1．Representation via SDEs

4.2．Representation via BSDEs

5．Forward-Backward Stochastic Differential Equations

5.1．General formulation and nonsolvability

5.2．The four-step scheme,a heuristie dcrivation

5.3．Several solvablc classes of FBSDEs

6．Option Pricing Problems

6.1．European call options and the Black-Scholes formula

6.2．Other options

7．Historical Remarks

References

Index