连续时间中的随机优化

List of Figures

Preface

1 Probability Theory

1.1 Introduction

1.2 Stochastic Processes

1.2.1 In formation Sets and a -Algebras

1.2.2 The Cantor Set

1.2.3 Borel-Cantelli Lemmas

1.2.4 Distribution Functions and Stochastic Processes

1.3 Conditional Expectation

1.3.1 Conditional Probability

1.3.2 Conditional Expectation

1.3,3 Change of Variables

1.4 Notes and Further Readings

2 Wiener Processes

2.1 introduction

2.2 A Heuristic Approach

2.2.1 From Random Walks to Wiener Process

2.2.2 Some Basic Properties of the Wiener Process

2.3 Markov Processes

2.3.1 Introduction

2.3.2 Transition Probability

2.3.3 Diffusion Processes

2.4 Wiener Processes

2.4.1 How to Generate More Wiener Processes

2.4.2 Differentiability of Sample Functions

2.4.3 Stopping Times

2.4.4 The Zero Set

2.4.5 Bounded Variations and the Irregularity of the

Wiener Process

2.5 Notes and Further Readings

3 Stochastic Calculus

3.1 Introduction

3.2 A Heuristic Approach

3.2.1 ls □ （s X ）dWs Riemarm Integrable?

3.2.2 The Choice of□ Matters

3.2.3 In Search of the Class of Functions for a （s, w）

3.3 The Ito Integral

3.3.1 Definition

3.3.2 Martingales

3.4 lto's Lemma: Autonomous Case

3.4.1 Ito's Lemma

3.4.2 Geometric Brownian Motion

3.4.3 Population Dynamics

3.4.4 Additive Shocks or Multiplicative Shocks

3.4.5 Multiple Sources of Uncertainty

3.4.6 Multivariate lto's Lemma

3.5 Ito's Lemma for Time-Dependent Functions

3.5.1 Euler's Homogeneous Differential Equation and the Heat Equation

3.5.2 Black-Scholes Formula

3.5.3 Irreversible Investment

3.5.4 Budget Equation for an Investor

3.5.5 Ito's Lemma: General Form

3.6 Notes and Further Readings

4 Stochastic Dynamic Programming

4.1 Introduction

4.2 Bellman Equation

4.2.1 Infinite-Horizon Problems

4.2.2 Verification Theorem

4.2.3 Finite-Horizon Problems

4.2.4 Existence and Differentiability of the Value Function

4.3 Economic Applications

4.3.1 Consumption and Portfolio Rules

4.3.2 Index Bonds

4.3.3 Exhaustible Resources

4.3.4 Adjustment Costs and （Reversible） Investment

4.3.5 Uncertain Lifetimes and Life Insurance

4.4 Extension: Reeursive Utility

4.4.1 Bellman Equation with Recursive Utility

4.4.2 Effects of Reeursivity: Deterministic Case

4.5 Notes and Further Readings

5 How to Solve it

5.1 Introduction

5.2 HARA Functions

5.2.1 The Meaning of Each Parameter

5.2.2 Closed-Form Representations

5.3 Trial and Error

5.3.1 Linear-Quadratic Models

5.3.2 Linear-HARA models

5.3.3 Linear-Concave Models

5.3,4 Nonlinear-Concave Models

5.4 Symmetry

5.4.1 Linear-Quadratic Model Revisited

5.4.2 Merton's Model Revisited

5.4.3 Fischer's Index Bond Model

5.4.4 Life Insurance

5.5 The Substitution Method

5.6 Martingale Representation Method

5.6.1 Girsanov Transformation

5.6.2 Example: A Portfolio Problem

5.6.3 Which 8 to Choose?

5.6.4 A Transformed Problem

5.7 Inverse Optimum Method

5.7.1 The Inverse Optimal Problem: Certainty Case

5.7.2 The Inverse Optimal Problem: Stochastic Case

5.7.3 Inverse Optimal Problem of Merton's Model

5.8 Notes and Further Readings

6 Boundaries and Absorbing Barriers

6.1 Introduction

6.2 Nonnegativity Constraint

6.2.1 Issues and Problems

6.2.2 Comparison Theorems

6.2.3 Chang and Malliaris's Reflection Method

6.2.4 Inaccessible Boundaries

6.3 Other Constraints

6.3.1 A Portfolio Problem with Borrowing CoosWaints

6.3.2 Viscosity Solutions

6.4 Stopping Rules - Certainty Case

6.4.1 The Baumol-Tobin Model

6.4.2 A Dynamic Model of Money Demand

6.4.3 The Tree-Cutting Problem

6.5 The Expected Discount Factor

6.5.1 Fundamental Equation for Ex[e□]

6.5.2 One Absorbing Barrier

6.5.3 Two Absorbing Barriers

6.6 Optimal Stopping Times

6.6.1 Dynamic and Stochastic Demand for Money

6.6.2 Stochastic Tree-Cutting and Rotation Problems

6.6.3 Investment Timing

6.7 Notes and Further Readings

A Miscellaneous Applications and Exercises

Bibliography

Index

"Stochastic optimization in continuous time"（AuthorFwu-Ranq Chang）is a rigorous but user-friendly book on the application of stochastic control theory to economics. A distinctive feature of the book is that math-ematical concepts are introduced in a language and terminology familiar to graduate students of economics.