The Art of Smooth Pasting, Vol. 55 Book

	Introduction to the Series
	Preface
1	Brownian Motion	1
1.1	Random Walk Representation	2
1.2	Ito's Lemma	4
1.3	Geometric Brownian Motion	6
1.4	Some Generalizations	8
2	Discounted Present Values	9
2.1	Present Values for Exponential and Polynomials	10
2.2	Present Values for Powers of Geometric Brownian Motion	13
2.3	A Basic Differential Equation for Present Value	14
2.4	Derivation by Discrete Approximation	15
2.3	The General Solution	16
2.6	Differential Equation for Geometric Brownian Motion	19
2.7	General Diffusion Processes	21
3	Barriers	22
3.1	The Basic Differential Equation	23
3.2	Geometric Brownian Motion	25
3.3	Stopping	25
3.4	Resetting	26
3.5	Reflection	26
3.6	Example: Price Ceiling	27
3.7	Example: Exchange Rate Target Zones	28
3.8	Transitional Boundary	30
3.9	Example: Temporary Suspension	31
4	Optimal Control and Regulation	32
4.1	Stopping	34
4.2	Example: Irreversible Investment	37
4.3	Convex Costs: Continuous Control	38
4.4	Lump-Sum Costs: Impulse Control	39
4.5	Example: Menu Costs	41
4.6	Linear Costs: Barrier Control	42
4.7	Some Geometry and Intuition	43
4.8	Example: Competitive Industry	45
5	Generalizations	47
5.1	Mean-Reverting Processes	47
5.2	Finite Horizon	49
6	Some Characterization of Optimal Paths	51
6.1	Short Run: Time Until First Action	52
6.2	Long Run: Stationary Distribution and Average Action	58
6.3	Dynamics of Brownian motion: Kolmogorov Equations	63
	References	69
	Index	71