Direct and to the point, this book from one of the field's leaders covers Brownian motion and stochastic calculus at the graduate level, and illustrates the use of that theory in various application domains, emphasizing business and economics. The mathematical development is narrowly focused and briskly paced, with many concrete calculations and a minimum of abstract notation. The applications discussed include: the role of reflected Brownian motion as a storage model, queueing model, or inventory model; optimal stopping problems for Brownian motion, including the influential McDonald-Siegel investment model; optimal control of Brownian motion via barrier policies, including optimal control of Brownian storage systems; and Brownian models of dynamic inference, also called Brownian learning models, or Brownian filtering models.

Preface ix

Guide to Notation and Terminology xv

1 Brownian Motion 1

1.1 Wiener’s theorem 1

1.2 Quadratic variation and local time 3

1.3 Strong Markov property 5

1 .4 Brownian martingales 6

1.5 Two characterizations of Brownian motion 7

1.6 The innovation theorem 7

1.7 A joint distribution (Reflection principle) 9

1.8 Change of drift as change of measure 10

1.9 A hjtting time distribution 13

1.10 Reflected Brownian motion 15

l.11 Problems and complements 16

2 Stochastic Storage Models 18

2.1 Buttered stochastic flow 19

2.2 The one-sided reflection mapping 20

2.3 Finjte butter capacity 22

2.4 The two-sided reflection mappi ng 23

2.5 Measuring system perform ance 25

2.6 Brownian storage models 30

2.7 Problems and complements 31

3 Further Analysis of Brownian Motion 36

3.1 Introduction 36

3.2 The backward and forward equations 37

3.3 Hitting time problems 38

3.4 Expected discounted costs 43

3.5 One absorbing barrier 44

3.7 More on reflected Browni an motion 48

3.8 Problems and complements 48

4 Stochastic Calculus 52

4.1 Introduction 52

4.2 First definition of the stochastic integral 53

4.3 An illu minating example 56

4.4 Final definition of the integral 58

4.5 Stochastic differential equations 60

4.6 Simplest version of Ito’s formula 61

4.7 The multi-dimen sional Ito formula 64

4.8 Tanaka’s formula and local time 65

4.9 Another generalization of Ito’s formula 68

4.10 Integration by parts (Special cases) 69

4.11 Differential equations for Brownian motion 70

4.12 Problems and complements 72

5 Optimal Stopping of Brownian Motion 77

5.1 A general problem 78

5.2 Continuation costs 79

5.3 McDonald-Siegel investment model 80

5.4 An investment problem with costly waiting 83

5.5 Some general theory 85

5.6 Sources and literature 88

5.7 Problems and complements 88

6 Reflected Brownian Motion 90

6.1 Strong Markov property 90

6.2 App]jcation of Ito’s formula 92

6.3 Expected discounted costs 93

6.4 Regenerative structure 95

6.5 The steady-state distribution 98

6.6 The case of a single barrier I OI

6.7 Problems and complements 103

7 Optimal Control of Brownian Motion 109

7.1 Impulse control wi th discounting 1 10

7.2 Control band policies 113

7.3 Optimal policy parameters 1 15

7.4 Impulse control with average cost criterion 120

7.5 Relative cost functions 121

7.6 Average cost optimality 123

7.7 Instantaneous control with discounting 126

7.8 Instantaneous control with average cost criterion 130

7.9 Cash management 133

7.10 Sources and literature 134

7.11 Problems and complements 135

8 Brownian Models of Dynamic Inference 137

8.1 Drift-rate uncertai nty in a Bayesian framework 138

8.2 Binary prior distributi on 138

8.3 Brownian sequential detection 141

8.4 General fi nite prior distribution 145

8.5 Gaussian prior distribution 148

8.6 A partially observed Markov chain 149

8.7 Change-point detection 152

9 Further Examples 155

9.1 Ornstein-Uhlenbeck process 155

9.2 Probability of ruin with compounding assets 156

9.3 RBM with killing at the boundary 159

9.4 Brownian motion with an interval removed 161

9.5 Brownian motion with sticky reflection 164

9.6 A capacity expansion model 166

9.7 Problems and complements 168

Appendix A Stochastic Processes 171

A.I Afiltered probability space 171

A.2 Random variables and stochastic processes 173

A.3 A canonicalexample 176

A.4 Two martingale theorems 176

A.5 A version of Fubini’s theorem 177

Appendix B Real Analysis 179

B.1 Absolutely continuous functions 179

B.2 VF functions 180

B.3 Memann-Stieltjes integration 180

B.4 The Riemann-Stieltjes chain rule 181

B.5 Notational conventions for integrals 182

References 183

Index 187

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