Brownian Motion and Stochastic Calculus

Front Cover
Springer Science & Business Media, 1991 - Mathematics - 470 pages
11 Reviews

This book is designed as a text for graduate courses in stochastic processes. It is written for readers familiar with measure-theoretic probability and discrete-time processes who wish to explore stochastic processes in continuous time. The vehicle chosen for this exposition is Brownian motion, which is presented as the canonical example of both a martingale and a Markov process with continuous paths. In this context, the theory of stochastic integration and stochastic calculus is developed. The power of this calculus is illustrated by results concerning representations of martingales and change of measure on Wiener space, and these in turn permit a presentation of recent advances in financial economics (option pricing and consumption/investment optimization).

This book contains a detailed discussion of weak and strong solutions of stochastic differential equations and a study of local time for semimartingales, with special emphasis on the theory of Brownian local time. The text is complemented by a large number of problems and exercises.

  

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it's very hard to read. you need to learn measure theory first

Contents

Martingales Stopping Times and Filtrations
1
12 Stopping Times
6
13 ContinuousTime Martingales
11
A Fundamental Inequalities
12
B Convergence Results
17
C The Optional Sampling Theorem
19
14 The DoobMeyer Decomposition
21
15 Continuous SquareIntegrable Martingales
30
A The MeanValue Property
241
B The Dirichlet Problem
243
C Conditions for Regularity
247
D Integral Formulas of Poisson
251
E Supplementary Exercises
253
43 The OneDimensional Heat Equation
254
A The Tychonoff Uniqueness Theorem
255
B Nonnegative Solutions of the Heat Equation
256

16 Solutions to Selected Problems
38
17 Notes
45
Brownian Motion
47
B The KolmogorovCentsov Theorem
53
23 Second Construction of Brownian Motion
56
24 The Space C0 oo Weak Convergence and the Wiener Measure
59
A Weak Convergence
60
B Tightness
61
C Convergence of FiniteDimensional Distributions
64
D The Invariance Principle and the Wiener Measure
66
25 The Markov Property
71
A Brownian Motion in Several Dimensions
72
B Markov Processes and Markov Families
74
C Equivalent Formulations of the Markov Property
75
26 The Strong Markov Property and the Reflection Principle
79
B Strong Markov Processes and Families
81
C The Strong Markov Property for Brownian Motion
84
27 Brownian Filtrations
89
A RightContinuity of the Augmented Filtration for a Strong Markov Process
90
B A Universal Filtration
93
C The Blumenthal ZeroOne Law
94
A Brownian Motion and Its Running Maximum
95
B Brownian Motion on a HalfLine
97
D Distributions Involving Last Exit Times
100
29 The Brownian Sample Paths
103
B The Zero Set and the Quadratic Variation
104
C Local Maxima and Points of Increase
106
D Nowhere Differentiability
109
E Law of the Iterated Logarithm
111
F Modulus of Continuity1
114
210 Solutions to Selected Problems
116
211 Notes
126
Stochastic Integration
128
32 Construction of the Stochastic Integral
129
A Simple Processes and Approximations
132
B Construction and Elementary Properties of the Integral
137
C A Characterization of the Integral
141
D Integration with Respect to Continuous Local Martingales
145
33 The ChangeofVariable Formula
148
A The Ito Rule
149
B Martingale Characterization of Brownian Motion
156
C Bessel Processes Questions of Recurrence
158
D Martingale Moment Inequalities
163
E Supplementary Exercises
167
34 Representations of Continuous Martingales in Terms of Brownian Motion
169
A Continuous Local Martingales as Stochastic Integrals with Respect to Brownian Motion
170
B Continuous Local Martingales as TimeChanged Brownian Motions
173
C A Theorem of F B Knight
179
D Brownian Martingales as Stochastic Integrals
180
E Brownian Functionals as Stochastic Integrals
185
35 The Girsanov Theorem
190
A The Basic Result
191
B Proof and Ramifications
193
C Brownian Motion with Drift
196
D The Novikov Condition
198
36 Local Time and a Generalized Ito Rule for Brownian Motion
201
A Definition of Local Time and the Tanaka Formula
203
B The Trotter Existence Theorem
206
C Reflected Brownian Motion and the Skorohod Equation
210
D A Generalized Ito Rule for Convex Functions
212
E The EngelbertSchmidt ZeroOne Law
215
37 Local Time for Continuous Semimartingales1
217
38 Solutions to Selected Problems
226
39 Notes
236
Brownian Motion and Partial Differential Equations
239
42 Harmonic Functions and the Dirichlet Problem
240
C BoundaryCrossing Probabilities for Brownian Motion
262
D Mixed InitialBoundary Value Problems
265
44 The Formulas of Feynman and Kac
267
A The Multidimensional Formula
268
B The OneDimensional Formula
271
45 Solutions to Selected Problems
276
46 Notes
278
Stochastic Differential Equations
281
52 Strong Solutions
284
A Definitions
285
B The ltd Theory
286
C Comparison Results and Other Refinements
291
D Approximations of Stochastic Differential Equations
295
E Supplementary Exercises
299
53 Weak Solutions
300
A Two Notions of Uniqueness
301
B Weak Solutions by Means of the Girsanov Theorem
302
C A Digression on Regular Conditional Probabilities
306
D Results of Yamada and Watanabe on Weak and Strong Solutions
308
54 The Martingale Problem of Stroock and Varadhan
311
A Some Fundamental Martingales
312
B Weak Solutions and Martingale Problems
314
C WellPosedness and the Strong Markov Property
319
D Questions of Existence
323
E Questions of Uniqueness
325
F Supplementary Exercises
328
55 A Study of the OneDimensional Case
329
A The Method of TimeChange
330
B The Method of Removal of Drift
339
C Fellers Test for Explosions
342
D Supplementary Exercises
351
56 Linear Equations
354
A GaussMarkov Processes
355
B Brownian Bridge
358
C The General OneDimensional Linear Equation
360
D Supplementary Exercises
361
57 Connections with Partial Differential Equations
363
A The Dirichlet Problem
364
B The Cauchy Problem and a FeynmanKac Representation
366
C Supplementary Exercises
369
58 Applications to Economics
371
B Option Pricing
376
C Optimal Consumption and Investment General Theory
379
D Optimal Consumption and Investment Constant Coefficients
381
59 Solutions to Selected Problems
387
510 Notes
394
P Levys Theory of Brownian Local Time
399
62 Alternate Representations of Brownian Local Time
400
B Poisson Random Measures
403
C Subordinators
405
D The Process of Passage Times Revisited
411
E The Excursion and Downcrossing Representations of Local Time
414
63 Two Independent Reflected Brownian Motions
418
B The First Formula of D Williams
421
C The Joint Density of Wt Lt r+t
423
64 Elastic Brownian Motion
425
A The FeynmanKac Formulas for Elastic Brownian Motion Our intent is to study the counterpart
426
B The RayKnight Description of Local Time
430
C The Second Formula of D Williams
434
Transition Probabilities of Brownian Motion with TwoValued Drift
437
66 Solutions to Selected Problems
442
67 Notes
445
Bibliography
447
Index
459
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About the author (1991)

Steven E. Shreve is Co-Founder of the Carnegie Mellon MS Program in Computational Finance and winner of the Carnegie Mellon Doherty Prize for sustained contributions to education.

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