## Stochastic Calculus and Financial ApplicationsThe 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 |

Itô Integration | 79 |

Itôs Integral as a Process | 82 |

Benefits and Costs | 85 |

65 Pathwise Interpretation of Ito Integrals | 87 |

66 Approximation in H² | 90 |

67 Exercises | 93 |

Localization and Itôs Integral | 95 |

72 An Intuitive Representation | 99 |

73 Why Just L²LOC ? | 102 |

74 Local Martingales and Honest Ones | 103 |

75 Alternative Fields and Changes of Time | 106 |

76 Exercises | 109 |

Itôs Formula | 111 |

82 First Consequences and Enhancements | 115 |

83 Vector Extension and Harmonic Functions | 120 |

84 Functions of Processes | 123 |

85 The General Itô 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 Lévys 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 |

293 | |

297 | |

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### Common terms and phrases

apply approximation arbitrage argument basic Black-Scholes bond bounded Brownian motion calculation called chapter coefficients complete condition consequence consider constant construction continuous convergence defined definition denote derivation diffusion distribution easy equal equation example EXERCISE existence expectation fact Finally follows formula function Gaussian given gives idea identity immediate important independent inequality integral interesting Itô integral lemma limit linear look martingale mean measure method natural observation option paths portfolio probability problem proof Proposition prove provides random variables random walk reason representation result satisfies sequence side simple solution solve space standard step stochastic stopping strategies Suppose tells theorem theory transform uniqueness write zero