Table of Contents
Preface xi
1 Preliminaries From Calculus 1
1.1 Functions in Calculus 1
1.2 Variation of a Function 4
1.3 Riemann Integral and Stieltjes Integral 9
1.4 Lebesgue's Method of Integration 14
1.5 Differentials and Integrals 14
1.6 Taylor's Formula and Other Results 15
2 Concepts of Probability Theory 21
2.1 Discrete Probability Model 21
2.2 Continuous Probability Model 28
2.3 Expectation and Lebesgue Integral 33
2.4 Transforms and Convergence 37
2.5 Independence and Covariance 39
2.6 Normal (Gaussian) Distributions 41
2.7 Conditional Expectation 43
2.8 Stochastic Processes in Continuous Time 47
3 Basic Stochastic Processes 55
3.1 Brownian Motion 56
3.2 Properties of Brownian Motion Paths 63
3.3 Three Martingales of Brownian Motion 65
3.4 Markov Property of Brownian Motion 67
3.5 Hitting Times and Exit Times 69
3.6 Maximum and Minimum of Brownian Motion 71
3.7 Distribution of Hitting Times 73
3.8 Reflection Principle and Joint Distributions 74
3.9 Zeros of Brownian Motion - Arcsine Law 75
3.10 Size of Increments of Brownian Motion 78
3.11 Brownian Motion in Higher Dimensions 81
3.12 Random Walk 81
3.13 Stochastic Integral in Discrete Time 83
3.14 Poisson Process 86
3.15 Exercises 88
4 Brownian Motion Calculus 91
4.1 Definition of Itô Integral 91
4.2 Itô Integral Process 100
4.3 Itô Integral and Gaussian Processes 103
4.4 Itô's Formula for Brownian Motion 106
4.5 Itô Processes and Stochastic Differentials 108
4.6 Itô's Formula for Ito Processes 112
4.7 Itô Processes in Higher Dimensions 118
4.8 Exercises 121
5 Stochastic Differential Equations 123
5.1 Definition of Stochastic Differential Equations (SDEs) 123
5.2 Stochastic Exponential and Logarithm 129
5.3 Solutions to Linear SDEs 131
5.4 Existence and Uniqueness of Strong Solutions 134
5.5 Markov Property of Solutions 136
5.6 Weak Solutions to SDEs 137
5.7 Construction of Weak Solutions 139
5.8 Backward and Forward Equations 144
5.9 Stratonovich Stochastic Calculus 146
5.10 Exercises 148
6 Diffusion Processes 151
6.1 Martingales and Dynkin's Formula 151
6.2 Calculation of Expectations and PDEs 155
6.3 Time-Homogeneous Diffusions 159
6.4 Exit Times from an Interval 163
6.5 Representation of Solutions of ODES 167
6.6 Explosion 168
6.7 Recurrence and Transience 170
6.8 Diffusion on an Interval 171
6.9 Stationary Distributions 172
6.10 Multi-dimensional SDEs 175
6.11 Exercises 183
7 Martingales 185
7.1 Definitions 185
7.2 Uniform Integrability 187
7.3 Martingale Convergence 189
7.4 Optional Stopping 191
7.5 Localization and Local Martingales 197
7.6 Quadratic Variation of Martingales 200
7.7 Martingale Inequalities 203
7.8 Continuous Martingales - Change of Time 205
7.9 Exercises 211
8 Calculus For Semimartingales 213
8.1 Semimartingales 213
8.2 Predictable Processes 214
8.3 Doob-Meyer Decomposition 215
8.4 Integrals with Respect to Semimartingales 217
8.5 Quadratic Variation and Covariation 220
8.6 Ito's Formula for Continuous Semimartingales 222
8.7 Local Times 224
8.8 Stochastic Exponential 226
8.9 Compensators and Sharp Bracket Process 230
8.10 Ito's Formula for Semimartingales 236
8.11 Stochastic Exponential and Logarithm 238
8.12 Martingale (Predictable) Representations 239
8.13 Elements of the General Theory 242
8.14 Random Measures and Canonical Decomposition 246
8.15 Exercises 249
9 Pure Jump Processes 251
9.1 Definitions 251
9.2 Pure Jump Process Filtration 252
9.3 Itô's Formula for Processes of Finite Variation 253
9.4 Counting Processes 254
9.5 Markov Jump Processes 261
9.6 Stochastic Equation for Jump Processes 264
9.7 Generators and Dynkin's Formula 265
9.8 Explosions in Markov Jump Processes 267
9.9 Exercises 268
10 Change of Probability Measure 269
10.1 Change of Measure for Random Variables 269
10.2 Change of Measure on a General Space 273
10.3 Change of Measure for Processes 276
10.4 Change of Wiener Measure 281
10.5 Change of Measure for Point Processes 283
10.6 Likelihood Functions 284
10.7 Exercises 287
11 Applications in Finance: Stock and FX Options 289
11.1 Financial Derivatives and Arbitrage 289
11.2 A Finite Market Model 295
11.3 Semimartingale Market Model 299
11.4 Diffusion and the Black-Scholes Model 304
11.5 Change of Numeraire 312
11.6 Currency (FX) Options 315
11.7 Asian, Lookback, and Barrier Options 318
11.8 Exercises 321
12 Applications in Finance: Bonds, Rates, and Options 325
12.1 Bonds and the Yield Curve 325
12.2 Models Adapted to Brownian Motion 327
12.3 Models Based on the Spot Rate 328
12.4 Merton's Model and Vasicek's Model 329
12.5 Heath-Jarrow-Morton (HJM) Model 333
12.6 Forward Measures - Bond as a Numeraire 338
12.7 Options, Caps, and Floors 341
12.8 Brace-Gatarek-Musiela (BGM) Model 343
12.9 Swaps and Swaptions 347
12.10 Exercises 349
13 Applications in Biology 353
13.1 Feller's Branching Diffusion 353
13.2 Wright-Fisher Diffusion 357
13.3 Birth-Death Processes 359
13.4 Growth of Birth-Death Processes 363
13.5 Extinction, Probability, and Time to Exit 366
13.6 Processes in Genetics 369
13.7 Birth-Death Processes in Many Dimensions 375
13.8 Cancer Models 377
13.9 Branching Processes 379
13.10 Stochastic Lotka-Volterra Model 386
13.11 Exercises 393
14 Applications in Engineering and Physics 395
14.1 Filtering 395
14.2 Random Oscillators 402
14.3 Exercises 408
Solutions to Selected Exercises 411
References 429
Index 435
