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Preface | |
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Introduction | |
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Notes | |
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Mathematical Preliminaries | |
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Stochastic Processes, Brownian Motions, and Diffusions | |
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Random Variables and Stochastic Processes | |
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Independence | |
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Wiener Processes and Brownian Motions | |
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Random Walk Approximation of a Brownian Motion | |
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Stopping Times | |
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Strong Markov Property | |
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Diffusions | |
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Discrete Approximation of an Ornstein-Uhlenbeck Process | |
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Notes | |
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Stochastic Integrals and Ito's Lemma | |
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The Hamilton-Jacobi-Bellman Equation | |
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Stochastic Integrals | |
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Ito's Lemma | |
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Geometric Brownian Motion | |
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Occupancy Measure and Local Time | |
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Tanaka's Formula | |
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The Kolmogorov Backward Equation | |
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The Kolmogorov Forward Equation | |
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Notes | |
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Martingales | |
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Definition and Examples | |
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Martingales Based on Eigenvalues | |
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The Wald Martingale | |
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Sub- and Supermartingales | |
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Optional Stopping Theorem | |
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Optional Stopping Theorem, Extended | |
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Martingale Convergence Theorem | |
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Notes | |
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Useful Formulas for Brownian Motions | |
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Stopping Times Defined by Thresholds | |
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Expected Values for Wald Martingales | |
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The Functions [psi] and [Psi] | |
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ODEs for Brownian Motions | |
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Solutions for Brownian Motions When r = 0 | |
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Solutions for Brownian Motions When r > 0 | |
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ODEs for Diffusions | |
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Solutions for Diffusions When r = 0 | |
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Solutions for Diffusions When r > 0 | |
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Notes | |
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Impulse Control Models | |
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Exercising an Option | |
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The Deterministic Problem | |
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The Stochastic Problem: A Direct Approach | |
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Using the Hamilton-Jacobi-Bellman Equation | |
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An Example | |
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Notes | |
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Models with Fixed Costs | |
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A Menu Cost Model | |
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Preliminary Results | |
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Optimizing: A Direct Approach | |
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Using the Hamilton-Jacobi-Bellman Equation | |
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Random Opportunities for Costless Adjustment | |
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An Example | |
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Notes | |
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Models with Fixed and Variable Costs | |
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An Inventory Model | |
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Preliminary Results | |
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Optimizing: A Direct Approach | |
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Using the Hamilton-Jacobi-Bellman Equation | |
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Long-Run Averages | |
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Examples | |
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Strictly Convex Adjustment Costs | |
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Notes | |
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Models with Continuous Control Variables | |
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Housing and Portfolio Choice with No Transaction Cost | |
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The Model with Transaction Costs | |
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Using the Hamilton-Jacobi-Bellman Equation | |
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Extensions | |
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Notes | |
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Instantaneous Control Models | |
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Regulated Brownian Motion | |
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One- and Two-Sided Regulators | |
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Discounted Values | |
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The Stationary Distribution | |
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An Inventory Example | |
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Notes | |
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Investment: Linear and Convex Adjustment Costs | |
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Investment with Linear Costs | |
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Investment with Convex Adjustment Costs | |
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Some Special Cases | |
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Irreversible Investment | |
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Irreversible Investment with Two Shocks | |
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A Two-Sector Economy | |
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Notes | |
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Aggregation | |
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An Aggregate Model with Fixed Costs | |
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The Economic Environment | |
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An Economy with Monetary Neutrality | |
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An Economy with a Phillips Curve | |
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Optimizing Behavior and the Phillips Curve | |
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Motivating the Loss Function | |
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Notes | |
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Continuous Stochastic Processes | |
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Modes of Convergence | |
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Continuous Stochastic Processes | |
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Wiener Measure | |
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Nondifferentiability of Sample Paths | |
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Notes | |
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Optional Stopping Theorem | |
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Stopping with a Uniform Bound, T < N | |
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Stopping with Pr { T < [infinity]} = 1 | |
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Notes | |
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References | |
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Index | |