Brownian Motion and Stochastic CalculusThis book is designed as a text for graduate courses in stochastic processes. It is written for readers familiar with measuretheoretic probability and discretetime 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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A great asset
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 
447  
459  
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Brownian Motion and Stochastic Calculus Ioannis Karatzas,J. Karatzas,Steven E. Shreve Snippet view  1988 