Advanced Mathematical Methods in Environmental and Resource Economics

The following sections are included:

• Preface
• About the Author
• Contents

An ordinary differential equation (ODE) of order n ≥ 1 can be informally defined as an equation involving an unknown function of one independent variable, which in most cases is taken to represent time, x(t), and its derivatives up to order n. The nth order differential equation can be written as

If x ∈ ℝⁿ, then (1) represents a system of n differential equations…

The following sections are included:

• Equilibrium
• Stability Concepts: Definitions
• Stability of Linear Systems
• Classification of Equilibrium Points (n = 2)
• Stability of Nonlinear Systems: Qualitative Analysis
• Asymptotic Linear Stability and Perturbations
• Manifold Theorems
• Stable Manifold and Topological Equivalence
• Global Stability (Lyapunov)
• Phase Diagram Analysis: Use of Isoclines
• Periodic Solutions
• Existence of Limit Cycles on the Plane (Poincare Bendixson)
• Stability of Competitive Equilibrium: Description
• Cournot Stability
• Predator–Prey, Lotka–Volterra Systems

The following sections are included:

• Control in Optimization Problems
• Possible Objectives
• The General Control Problem
• Optimizing J
• Principle of Optimality
• Optimization in Stages
• A Shortest Distance Problem
• Solution
• State Structure Optimization and Dynamic Programming
• Backward Recursion for the Objective
• Solution Approaches
• Dynamic Programming
• Feedback Control
• Discounting and Time Independence

The following sections are included:

• The Problem
• The Euler–Lagrange Equation
• Forms of the Euler Equation
• Special Cases of the Euler Equation
• Canonical Form of the Euler Equation
• Second Order Conditions
• Transversality Conditions
• Salvage (Scrap) Value
• Optimal Extraction
• Several Unknown Functions: The Problem
• The Euler Equation (Necessary Conditions)

The following sections are included:

• The Problem
• The Pontryagin Maximum Principle
• Sufficient Conditions
• Several Unknown Functions: The Problem
• The Pontryagin Maximum Principle for Many State Variables
• Optimal Control with Discounted Objective: Current Value Hamiltonians
• Interpretation of the Costate Variables
• Open Final Time Problems
• Infinite Horizon Optimal Control
• Pontryagin’s Maximum Principle
• Transversality Conditions
• Solution in the Control-State Space
• References

The following sections are included:

• Infinite Horizon Optimal Control
• Steady State
• Properties of the Steady State
• Local Stability
• Stable Manifold
• Solving for the Linearization Around the Steady State: Linear Quadratic Optimal Control Problem
• Linear Stable Manifold — Linear Policy Function
• Nonlinear Stable Manifold
• Global Asymptotic Stability
• The Globally Stable Manifold
• The Ramsey Problem
• Social Planner
• Steady State Analysis
• The Decentralized Economy
• Firms Maximize Profits
• TVC and No-Ponzi-Game Condition

The following sections are included:

• Linear Optimal Control
• A More General Linear Problem
• Solution Steps
• The Fishery Problem
• The Optimal Investment Problem (No Adjustment Costs)
• Most Rapid Approach Paths
• Determining the Most Rapid Approach Path
• Growth with Linear Utility
• The Optimal MRAP Policy

The following sections are included:

• Mixed Control and State Constraints
• Maximum Principle under Mixed Constraints
• Pure State Constraints
• Maximum Principle under Pure State Constraints
• Optimal Investment with Adjustment Costs and Pure State Constraints

The following sections are included:

• The Optimal Value Function
• Derivation of the Hamilton–Jacobi–Bellman (HJB) or Dynamic Programming Equation
• Verification
• Specific Forms of the HJB Equation
• The HJB Equation for Infinite Horizon Time Autonomous Problems
• Dynamic Programming and Maximum Principle
• Optimal Growth

The following sections are included:

• Lagrange Multipliers Methods
• Dynamic Programming
• The Value Function
• The HJB or Dynamic Programming Equation
• Properties of the Value Function
• Methods for Solving the Optimal Control Problem
• The Savings Problem
• Solution by Lagrange Multipliers
• Solution by Dynamic Programming
• The Ramsey Problem
• The Linear Quadratic Regulator (LQR)
• The Matrix Riccati Equation
• The Stochastic Linear Quadratic Regulator (S-LQR)
• Certainty Equivalence Principle (CEP)

The following sections are included:

• Sigma — Algebras and Probability Measures
• Random Variables
• Distribution Functions
• Probability Density Functions
• Stochastic Processes
• Stationary Processes
• White Noise
• The Random Walk
• The Wiener Process
• The Markov Process
• The Poisson Process
• Stochastic Differential Equations
• Itô Stochastic Differential Equation
• Geometric Brownian Motion
• The Ornstein–Uhlenbeck Process Mean Reversing Process)
• Itô’s Lemma
• The Black–Scholes Formula
• Existence and Uniqueness of Solutions, Dependence on Parameters and Initial Conditions
• Reference

The following sections are included:

• Optimal Stochastic Control: The Problem
• The Objective
• The Hamilton–Jacobi–Bellman (HJB) or Dynamic Programming Equation
• HJB Finite Horizon
• The Verification Theorem
• Infinite Time Horizon
• Solving the Optimal Stochastic Control Problem
• Problems in Economics
• Optimal Portfolio Rules
• Viscosity Solutions and Solvability of the HJB Equation
• The Deterministic Problem
• Classical and Weak Solutions
• The Stochastic Problem
• Reference

The following sections are included:

• Description
• Control and Strategy Spaces
• Information Structures
• Strong and Weak Time Consistency
• Dynamic Programming Characterization of FBNE Solutions
• A Linear-Quadratic Differential Game
• Dynamic Programming Approach
• Cooperative and Noncooperative Solutions
• Dynamic Duopoly
• Reference

The following sections are included:

• International Pollution Control under Risk
• The Lake Game —Nonlinear Problems
• Regulation in Nonlinear Differential Games
• Concluding Remarks
• References

The following sections are included:

• Expected Utility
• Knightian Uncertainty
• Ambiguity
• Robust Control (Hansen and Sargent)
• Robustness
• Kullback–Leibler Divergence
• Preferences for Robustness
• Regulation Breakdown
• Solution
• References

The following sections are included:

• Introduction
• Local Effects
• Nonlocal Effects
• Turing Instability
• Optimal Instability
• Global Analysis
• Robust Control in Space
• Spatial Interactions
• The Problem
• Translation Invariance
• Hot Spots
• The General LQ Problem
• Robust Control of In-Situ Consumption
• References

The following sections are included:

• Selected Bibliography
• Index