Franklin

Stochastic Optimization in Continuous Time.

Author/Creator:
Chang, Fwu-Ranq.
Publication:
Cambridge : Cambridge University Press, 2004.
Format/Description:
Book
1 online resource (346 pages)
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Subjects:
Economics -- Mathematical models.
Form/Genre:
Electronic books.
Summary:
This is an introduction to stochastic control theory with applications to economics, first published in 2004.
Contents:
Cover
Half-title
Title
Copyright
Dedication
Contents
List of Figures
Preface
1 Probability Theory
1.1 Introduction
1.2 Stochastic Processes
1.2.1 Information Sets and σ-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 Is Sigma (s, Xs) dWs Riemann Integrable?
3.2.2 The Choice of Ti Matters
3.2.3 In Search of the Class of Functions for Sigma (s, ω)
3.3 The Ito Integral
3.3.1 Definition
3.3.2 Martingales
3.4 Ito'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 Ito's Lemma
3.5 Ito's Lemma for Time-Dependent Functions
3.5.1 Euler's Homogeneous Differential Equation and the Heat Equation
Euler's Homogenous Differential Equation
On the Heat Equation
3.5.2 Black-Scholes Formula
3.5.3 Irreversible Investment
The Value of Investment Opportunity.
The Value an Investment Project
3.5.4 Budget Equation for an Investor
3.5.5 Ito's Lemma: General Form
Real-Valued Functions
Vector-Valued Functions
3.6 Notes and Further Readings
4 Stochastic Dynamic Programming
4.1 Introduction
4.2 Bellman Equation
4.2.1 Infinite-Horizon Problems
An Optimal Growth Problem
Derivation of the Bellman Equation
General Formulation
Dynkin's Formula
4.2.2 Verification Theorem
The Solution to (4.16) is the Expected Utility
Verification Theorem
4.2.3 Finite-Horizon Problems
4.2.4 Existence and Differentiability of the Value Function
Admissible Controls
One-Dimensional Controlled Processes
Multidimensional Controlled Processes
4.3 Economic Applications
4.3.1 Consumption and Portfolio Rules
An Infinite-Horizon Model
Wealth Effect
Mutual Fund Theorem
4.3.2 Index Bonds
A Three-Asset Model
Optimal Portfolio Rules
Insurance Premium
4.3.3 Exhaustible Resources
Dasgupta and Heal (1974)
Stochastic Extension
The Model in Pindyck (1980)
4.3.4 Adjustment Costs and (Reversible) Investment
User Cost
Dynamic Model without Adjustment Costs
Adjustment Costs
The Model in Pindyck (1982)
4.3.5 Uncertain Lifetimes and Life Insurance
Survival Function
Is More Information Better? An Anecdote
Life Insurance
4.4 Extension: Recursive Utility
4.4.1 Bellman Equation with Recursive Utility
4.4.2 Effects of Recursivity: Deterministic Case
Costate Equation
Euler Equation
Phase Diagram
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
DARA and IRRA (a >
0, b >
0)
CARA (a = 0, b >
0) - Exponential Utility Functions
IARA and IRRA (a <
0, b >
0)
CRRA (a >.
0, b = 0)
DARA and DRRA (a >
0, b <
0)
5.3 Trial and Error
5.3.1 Linear-Quadratic Models
5.3.2 Linear-HARA models
CARA Utility Functions
CRRA Utility Functions
Logarithmic Utility Functions
5.3.3 Linear-Concave Models
Competitive Firm
How to Guess the Solution
Implications
5.3.4 Nonlinear-Concave Models
An Optimal Growth Model
How to Guess the Solution
5.4 Symmetry
5.4.1 Linear-Quadratic Model Revisited
5.4.2 Merton's Model Revisited
CRRA Utility Functions
CARA Utility Functions
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 Theta 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
Existence Theorem
5.7.3 Inverse Optimal Problem of Merton's Model
Classical Formulation
General Formulation
An Explicit Example
5.8 Notes and Further Readings
6 Boundaries and Absorbing Barriers
6.1 Introduction
6.2 Nonnegativity Constraint
6.2.1 Issues and Problems
Standard Approach
Some Useful Theorems
6.2.2 Comparison Theorems
Ordinary Differential Equation
Stochastic Differential Equation
6.2.3 Chang and Malliaris's Reflection Method
Extending (6.3) to the Whole Real Line
Existence and Uniqueness Theorem
6.2.4 Inaccessible Boundaries
6.3 Other Constraints
6.3.1 A Portfolio Problem with Borrowing Constraints
General Setting
6.3.2 Viscosity Solutions
6.4 Stopping Rules-Certainty Case
6.4.1 The Baumol-Tobin Model
Square-Root Rule
Optimal Scheduling
Integer Constraint?
6.4.2 A Dynamic Model of Money Demand
Optimal Scheduling.
Formulation of Money Demand
Properties of Money Demand
6.4.3 The Tree-Cutting Problem
Without Rotation
With Rotation
6.5 The Expected Discount Factor
6.5.1 Fundamental Equation for…
Wald Martingales
Optional Stopping Theorem
6.5.2 One Absorbing Barrier
One Absorbing Barrier: X (t) = 0
Expected Discounted Cost Up to a Stopping Time
6.5.3 Two Absorbing Barriers
Two Absorbing Barriers 0 and b, with…
6.6 Optimal Stopping Times
6.6.1 Dynamic and Stochastic Demand for Money
Absorbing Barrier
Formulation Demand-for-Money Function
Properties of the Money Demand
6.6.2 Stochastic Tree-Cutting and Rotation Problems
Without Rotation
With Rotation
6.6.3 Investment Timing
6.7 Notes and Further Readings
APPENDIX A Miscellaneous Applications and Exercises
1. FARMLAND INVESTMENT
2. FUTURES PRICING
3. HABIT FORMATION AND PORTFOLIO SELECTION
4. MONEY AND GROWTH
5. A STOCHASTICALLY GROWING MONETARY MODEL
6. GROWTH AND TRADE
7. INTEGRABILITY PROBLEM OF ASSET PRICES
8. ENTRY AND EXIT
9. INVESTMENT LAGS
10. BOND PRICE DYNAMICS
11. ROTATION PROBLEMS AND GEOMETRIC GROWTH
12. ENDOGENOUS SEEDLING COST
Bibliography
Index.
Notes:
Description based on publisher supplied metadata and other sources.
Local notes:
Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2021. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries.
Other format:
Print version: Chang, Fwu-Ranq Stochastic Optimization in Continuous Time
ISBN:
9780511193941
9780521834063
OCLC:
144618374