Stochastic Calculus and Financial Applications

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Springer Science & Business Media, 2001 - Business & Economics - 300 pages
The Wharton School course on which the book is based is designed for energetic students who have had some experience with probability and statistics, but who have not had advanced courses in stochastic processes. Even though the course assumes only a modest background, it moves quickly and - in the end - students can expect to have the tools that are deep enough and rich enough to be relied upon throughout their professional careers. The course begins with simple random walk and the analysis of gambling games. This material is used to motivate the theory of martingales, and, after reaching a decent level of confidence with discrete processes, the course takes up the more demanding development of continuous time stochastic process, especially Brownian motion. The construction of Brownian motion is given in detail, and enough material on the subtle properties of Brownian paths is developed so that the student should sense of when intuition can be trusted and when it cannot. The course then takes up the It┐ integral and aims to provide a development that is honest and complete without being pedantic. With the It┐ integral in hand, the course focuses more on models. Stochastic processes of importance in Finance and Economics are developed in concert with the tools of stochastic calculus that are needed in order to solve problems of practical importance. The financial notion of replication is developed, and the Black-Scholes PDE is derived by three different methods. The course then introduces enough of the theory of the diffusion equation to be able to solve the Black-Scholes PDE and prove the uniqueness of the solution.
 

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Contents

Random Walk and First Step Analysis
11
12 Time and Infinity
11
13 Tossing an Unfair Coin
11
14 Numerical Calculation and Intuition
11
16 Exercises
11
First Martingale Steps
11
22 New Martingales from Old
13
23 Revisiting the Old Ruins
15
92 OrnsteinUhlenbeck Processes
138
93 Matching Product Process Coefficients
139
94 Existence and Uniqueness Theorems
142
95 Systems of SDEs
148
96 Exercises
149
Arbitrage and SDEs
153
102 The BlackScholes Model
156
103 The BlackScholes Formula
158

24 Submartingales
17
25 Doobs Inequalities
19
26 Martingale Convergence
22
27 Exercises
26
Brownian Motion
29
31 Covariances and Characteristic Functions
30
32 Visions of a Series Approximation
33
33 Two Wavelets
35
34 Wavelet Representation of Brownian Motion
36
35 Scaling and Inverting Brownian Motion
40
36 Exercises
41
Martingales The Next Steps
43
42 Conditional Expectations
44
43 Uniform Integrability
47
44 Martingales in Continuous Time
50
45 Classic Brownian Motion Martingales
55
46 Exercises
58
Richness of Paths
61
52 Not Too Smooth
63
53 Two Reflection Principles
66
54 The Invariance Principle and Donskers Theorem
70
55 Random Walks Inside Brownian Motion
72
56 Exercises
77
It˘ Integration
79
It˘s Integral as a Process
82
Benefits and Costs
85
65 Pathwise Interpretation of Ito Integrals
87
66 Approximation in H▓
90
67 Exercises
93
Localization and It˘s Integral
95
72 An Intuitive Representation
99
73 Why Just L▓LOC ?
102
74 Local Martingales and Honest Ones
103
75 Alternative Fields and Changes of Time
106
76 Exercises
109
It˘s Formula
111
82 First Consequences and Enhancements
115
83 Vector Extension and Harmonic Functions
120
84 Functions of Processes
123
85 The General It˘ Formula
126
86 Quadratic Variation
128
87 Exercises
134
Stochastic Differential Equations
137
104 Two Original Derivations
160
105 The Perplexing Power of a Formula
165
106 Exercises
167
The Diffusion Equation
169
112 Solutions of the Diffusion Equation
172
113 Uniqueness of Solutions
178
114 How to Solve the BlackScholes PDE
182
115 Uniqueness and the BlackScholes PDE
187
116 Exercises
189
Representation Theorems
191
122 The Martingale Representation Theorem
196
123 Continuity of Conditional Expectations
201
124 Representation via Time Change
203
125 Levys Characterization of Brownian Motion
204
126 Bedrock Approximation Techniques
206
127 Exercises
211
Girsanov Theory
213
132 Tilting a Process
215
133 Simplest Girsanov Theorem
218
134 Creation of Martingales
221
135 Shifting the General Drift
222
136 Exponential Martingales and Novikovs Condition
225
137 Exercises
229
Arbitrage and Martingales
233
142 The Valuation Formula in Continuous Time
235
143 The BlackScholes Formula via Martingales
241
144 American Options
244
145 SelfFinancing and SelfDoubt
246
146 Admissible Strategies and Completeness
252
147 Perspective on Theory and Practice
257
148 Exercises
259
The FeynmanKac Connection
263
152 The FeynmanKac Connection for Brownian Motion
265
153 LÚvys Arcsin Law
267
154 The FeynmanKac Connection for Diffusions
270
155 FeynmanKac and the BlackScholes PDEs
271
156 Exercises
274
Mathematical Tools
277
Comments and Credits
285
Bibliography
293
Index
297
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