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　　"Stochastic optimization in continuous time"（AuthorFwu-RanqChang）is a rigorous but user-friendly book on the application ofstochastic control theory to economics. A distinctive feature ofthe book is that math-ematical concepts are introduced in alanguage and terminology familiar to graduate students ofeconomics.

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 Functi*** 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 Functi***

 2.4.3 Stopping Times

 2.4.4 The Zero Set

 2.4.5 Bounded Variati*** 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 Functi*** 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 Functi***

 3.5.1 Euler's Homogeneous Differential Equation and theHeat Equation

 3.5.2 Black-Sc***s 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 ValueFunction

 4.3 Economic Applicati***

 4.3.1 C***umption 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 Functi***

 5.2.1 The Meaning of Each Parameter

 5.2.2 Closed-Form Representati***

 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.*** 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 C***traint

 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 C***traints

 6.3.1 A Portfolio Problem with Borrowing CoosWaints

 6.3.2 Viscosity Soluti***

 *** Stopping Rules - Certainty Case

 ***.1 The Baumol-Tobin Model

 ***.2 A Dynamic Model of Money Demand

 ***.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 Applicati*** and Exercises

Bibliography

Index   

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