Art of Smooth Pasting

BROWNIAN MOTION; Random Walk Representation; lto's Lemma; Geometric Brownian Motion; Some Generalizations; DISCOUNTED PRESENT VALUES Present Values for Exponential and Polynomials Present Values for Powers of Geometric Brownian Motion; A Basic Differential Equation for Present Value Derivation by Discrete Approximation; The General Solution; Differential Equation for Geometric Brownian Motion; General Diffusion Processes; BARRIERS; The Basic Differential Equation; Geometric Brownian Motion; Stopping and Resetting; Reflection; Example: Price Ceiling; Example: Exchange Rate Target Zones; Transitional Boundary; Example: Temporary Suspension; OPTIMAL CONTROL AND REGULATION; Stopping; Example: Irreversible Investment; Convex Costs: Continuous Control; Lump-Sum Costs: Impulse Control; Example: Menu Costs; Linear Costs: Barrier Control; Some Geometry and Intuition; Example: Competitive Industry; GENERALIZATIONS; Mean-Reverting Processes; Finite Horizon; SOME CHARACTERIZATION OF OPTIMAL PATHS; Short Run: Time Until First Action; Long Run: Stationary Distribution and Average Action; Dynamics of Brownian Motion: Kolmogorov Equations.