Introduction to stochastic calculus with applications

Fima C. Klebner1860945554, 9781860945557, 9781860946837, 186094566X

This book presents a concise treatment of stochastic calculus and its applications. It gives a simple but rigorous treatment of the subject including a range of advanced topics, it is useful for practitioners who use advanced theoretical results. It covers advanced applications, such as models in mathematical finance, biology and engineering. Self-contained and unified in presentation, the book contains many solved examples and exercises. It may be used as a textbook by advanced undergraduates and graduate students in stochastic calculus and financial mathematics. It is also suitable for practitioners who wish to gain an understanding or working knowledge of the subject. For mathematicians, this book could be a first text on stochastic calculus; it is good companion to more advanced texts by a way of examples and exercises. For people from other fields, it provides a way to gain a working knowledge of stochastic calculus. It shows all readers the applications of stochastic calculus methods and takes readers to the technical level required in research and sophisticated modelling. This second edition contains a new chapter on bonds, interest rates and their options. New materials include more worked out examples in all chapters, best estimators, more results on change of time, change of measure, random measures, new results on exotic options, FX options, stochastic and implied volatility, models of the age-dependent branching process and the stochastic Lotka–Volterra model in biology, non-linear filtering in engineering and five new figures.

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Table of contents :
Cover Page……Page 1
Title Page……Page 4
ISBN 1860945554……Page 5
Preface to the First Edition……Page 6
Acknowledgments……Page 8
Contents (with page links)……Page 10
Continuous and Differentiable Functions……Page 16
Right and Left-Continuous Functions……Page 17
1.2 Variation of a Function……Page 19
Continuous and Discrete Parts of a Function……Page 22
Quadratic Variation……Page 23
Riemann Integral……Page 24
Stieltjes Integral with respect to Monotone Functions……Page 25
Particular Cases……Page 26
Integration by Parts……Page 27
Change of Variables……Page 28
1.5 Differentials and Integrals……Page 29
Taylor’s Formula for Functions of One Variable……Page 30
Taylor’s Formula for Functions of Several Variables……Page 31
Lipschitz and Holder Conditions……Page 32
Growth Conditions……Page 33
Further Results on Functions and Integration……Page 34
Filtered Probability Space……Page 36
Fields of Events……Page 37
Stochastic Processes……Page 38
Filtration Generated by a Stochastic Process……Page 39
Stopping Times……Page 40
Expectation……Page 41
Conditional Expectation……Page 42
-Fields……Page 43
Probability……Page 44
Random Variables……Page 45
Distribution of a Random Variable……Page 46
Transformation of Densities……Page 47
2.3 Expectation and Lebesgue Integral……Page 48
Lebesgue Integral on the Line……Page 49
Jumps and Probability Densities……Page 50
Decomposition of Distributions and FV Functions……Page 51
2.4 Transforms and Convergence……Page 52
Convergence of Random Variables……Page 53
Independence……Page 54
Covariance……Page 55
2.6 Normal (Gaussian) Distributions……Page 56
Conditional Expectation and Conditional Distribution……Page 58
Properties of Conditional Expectation……Page 59
2.8 Stochastic Processes in Continuous Time……Page 62
Continuity and Regularity of Paths……Page 63
Filtered Probability Space and Adapted Processes……Page 64
Martingales, Supermartingales, Submartingales……Page 65
Stopping Times……Page 66
Fubini’s Theorem……Page 68
Introduction……Page 70
Defining Properties of Brownian Motion……Page 71
Space Homogeneity……Page 73
Brownian Motion as a Gaussian Process……Page 74
Brownian Motion as a Random Series……Page 77
Quadratic Variation of Brownian Motion……Page 78
Properties of Brownian paths……Page 79
3.3 Three Martingales of Brownian Motion……Page 80
3.4 Markov Property of Brownian Motion……Page 82
Stopping Times and Strong Markov Property……Page 83
3.5 Hitting Times and Exit Times……Page 84
3.6 Maximum and Minimum of Brownian Motion……Page 86
3.7 Distribution of Hitting Times……Page 88
3.8 Reflection Principle and Joint Distributions……Page 89
3.9 Zeros of Brownian Motion. Arcsine Law……Page 90
3.10 Size of Increments of Brownian Motion……Page 93
Graphs of some Functions of Brownian Motion……Page 94
3.12 Random Walk……Page 96
Martingales in Random Walks……Page 97
3.13 Stochastic Integral in Discrete Time……Page 98
Stopped Martingales……Page 100
Defining Properties of Poisson process……Page 101
Poisson Process Martingales……Page 102
3.15 Exercises……Page 103
Ito Integral of Simple Processes……Page 106
Properties of the Ito Integral of Simple Adapted Processes……Page 108
Ito Integral of Adapted Processes……Page 110
Martingale Property of the Ito Integral……Page 115
Quadratic Variation and Covariation of Ito Integrals……Page 116
4.3 Ito Integral and Gaussian Processes……Page 118
4.4 Ito’s Formula for Brownian Motion……Page 120
Definition of Ito Processes……Page 123
Quadratic Variation of Ito Processes……Page 124
4.6 Ito’s Formula for Ito Processes……Page 126
Integration by Parts……Page 127
Ito’s Formula for Functions of Two Variables……Page 131
4.7 Ito Processes in Higher Dimensions……Page 132
Ito’s Formula for Functions of Several Variables……Page 134
4.8 Exercises……Page 135
Ordinary Differential Equations……Page 138
White Noise and SDEs……Page 139
A Physical Model of Diffusion and SDEs……Page 140
Stochastic Differential Equations……Page 141
5.2 Stochastic Exponential and Logarithm……Page 143
5.3 Solutions to Linear SDEs……Page 145
General Linear SDEs……Page 146
Brownian Bridge……Page 147
5.4 Existence and Uniqueness of Strong Solutions……Page 148
Less Stringent Conditions for Strong Solutions……Page 149
Transition Function…….Page 150
5.6 Weak Solutions to SDEs……Page 151
Canonical Space for Diffusions……Page 153
Probability Measure……Page 154
SDE on the Canonical Space is Satisfied……Page 155
Weak Solutions and the Martingale Problem……Page 156
5.8 Backward and Forward Equations……Page 158
Integration by Parts: Stratanovich Product rule……Page 160
Change of Variables: Stratanovich Chain rule……Page 161
5.10 Exercises……Page 162
6.1 Martingales and Dynkin’s Formula……Page 164
6.2 Calculation of Expectations and PDEs……Page 168
Feynman-Kac formula……Page 170
6.3 Time Homogeneous Diffusions……Page 171
Ito’s Formula and Martingales……Page 174
6.4 Exit Times from an Interval……Page 175
6.5 Representation of Solutions of ODEs……Page 180
6.6 Explosion……Page 181
6.7 Recurrence and Transience……Page 182
6.8 Diffusion on an Interval……Page 184
6.9 Stationary Distributions……Page 185
Invariant Measures……Page 187
6.10 Multi-Dimensional SDEs……Page 188
Bessel Process……Page 190
Ito’s Formula, Dynkin’s Formula……Page 191
Recurrence, Transience and Stationary Distributions……Page 193
Higher Order Random Differential Equations……Page 194
6.11 Exercises……Page 195
7.1 Definitions……Page 198
Square Integrable Martingales……Page 199
7.2 Uniform Integrability……Page 200
7.3 Martingale Convergence……Page 202
7.4 Optional Stopping……Page 204
Optional Stopping of Discrete Time Martingales……Page 206
Gambler’s Ruin……Page 207
Hitting Times in Random Walks……Page 208
7.5 Localization and Local Martingales……Page 210
Dirichlet Class (D)……Page 212
7.6 Quadratic Variation of Martingales……Page 213
7.7 Martingale Inequalities……Page 215
7.8 Continuous Martingales. Change of Time……Page 217
Levy’s Characterization of Brownian Motion……Page 218
Change of Time for Martingales……Page 219
Change of Time in SDEs……Page 220
7.9 Exercises……Page 224
8.1 Semimartingales……Page 226
8.2 Predictable Processes……Page 227
Doob’s Decomposition……Page 229
Stochastic Integral with respect to Martingales……Page 230
Stochastic Integrals with respect to Semimartingales……Page 231
Properties of Stochastic Integrals with respect to Semimartingales……Page 232
Properties of Quadratic Variation……Page 233
Quadratic Variation of Stochastic Integrals……Page 234
8.6 Ito’s Formula for Continuous Semimartingales……Page 235
Ito’s Formula for Functions of Several Variables……Page 236
8.7 Local Times……Page 237
8.8 Stochastic Exponential……Page 239
Stochastic Exponential of Martingales……Page 240
8.9 Compensators and Sharp Bracket Process……Page 243
Sharp Bracket for Square Integrable Martingales……Page 245
Continuous Martingale Component of a Semimartingale……Page 246
Conditions for Existence of a Stochastic Integral……Page 247
Properties of the Predictable Quadratic Variation……Page 248
8.10 Ito’s Formula for Semimartingales……Page 249
8.11 Stochastic Exponential and Logarithm……Page 251
8.12 Martingale (Predictable) Representations……Page 252
8.13 Elements of the General Theory……Page 255
Stochastic Sets……Page 256
Classification of Stopping Times……Page 257
Random Measure for a Single Jump……Page 259
Random Measure of Jumps and its Compensator in Discrete Time……Page 260
Random Measure of Jumps and its Compensator……Page 261
8.15 Exercises……Page 262
9.1 Definitions……Page 264
9.2 Pure Jump Process Filtration……Page 265
Stochastic Exponential……Page 266
9.4 Counting Processes……Page 267
Point Process of a Single Jump……Page 269
Stochastic Intensity……Page 270
Non-homogeneous Poisson Processes……Page 271
Compensators of Pure Jump Processes……Page 273
The Compensator and the Martingale……Page 274
9.6 Stochastic Equation for Jump Processes……Page 276
Generators and Dynkin’s Formula……Page 277
9.7 Explosions in Markov Jump Processes……Page 278
9.8 Exercises……Page 280
Change of Measure on a Discrete Probability Space……Page 282
Change of Measure for Normal Random Variables……Page 283
10.2 Change of Measure on a General Space……Page 286
10.3 Change of Measure for Processes……Page 289
Change of Drift in Diffusions……Page 293
10.4 Change of Wiener Measure……Page 294
10.5 Change of Measure for Point Processes……Page 295
Likelihood for Discrete Observations……Page 297
Likelihood Ratios for Diffusions……Page 298
10.7 Exercises……Page 300
11.1 Financial Derivatives and Arbitrage……Page 302
Arbitrage and Fair Price……Page 304
Equivalence Portfolio. Pricing by No Arbitrage……Page 305
Binomial Model……Page 306
Pricing by No Arbitrage……Page 307
11.2 A Finite Market Model……Page 308
Arbitrage in Continuous Time Models……Page 312
EMM Assumption……Page 314
Admissible Strategies……Page 315
Pricing of Claims……Page 316
11.4 Diffusion and the Black-Scholes Model……Page 317
Black-Scholes Model……Page 318
Pricing a Call Option……Page 319
Pricing of Claims by a PDE. Replicating Portfolio……Page 320
Stochastic Volatility Models……Page 322
11.5 Change of Numeraire……Page 325
SDEs under a Change of Numeraire……Page 326
11.6 Currency (FX) Options……Page 327
Options on Foreign Assets Struck in Foreign Currency……Page 329
Asian Options……Page 330
Lookback Options……Page 332
Barrier Options……Page 333
11.8 Exercises……Page 334
12.1 Bonds and the Yield Curve……Page 338
EMM Assumption……Page 339
12.2 Models Adapted to Brownian Motion……Page 340
12.3 Models Based on the Spot Rate……Page 341
Merton’s Model……Page 342
Vasicek’s Model……Page 343
Bonds and Rates under Q and the No-arbitrage Condition……Page 346
Forward Measures……Page 351
Distributions of the Bond in HJM with Deterministic Volatilities……Page 352
Cap and Caplets……Page 354
Caplet Pricing in HJM model……Page 355
LIBOR……Page 356
SDEs for Forward LIBOR under Different Measures……Page 358
12.9 Swaps and Swaptions……Page 360
12.10 Exercises……Page 362
13.1 Feller’s Branching Diffusion……Page 366
13.2 Wright-Fisher Diffusion……Page 369
13.3 Birth-Death Processes……Page 371
Birth-Death Processes with Linear Rates……Page 372
Processes with Stabilizing Reproduction……Page 374
13.4 Branching Processes……Page 375
Deterministic Lotka-Volterra system……Page 381
Stochastic Lotka-Volterra system……Page 382
Existence……Page 383
Semimartingale decomposition for (xtK, ytK)……Page 385
Deterministic (Fluid) approximation……Page 386
13.6 Exercises……Page 388
14.1 Filtering……Page 390
General Non-linear Filtering Model……Page 391
Kalman-Bucy Filter……Page 394
14.2 Random Oscillators……Page 397
Non-linear Systems……Page 399
A System with a Cylindric Phase Plane……Page 400
14.3 Exercises……Page 403
Solutions to Selected Exercises……Page 406
References……Page 422
Index (with page links)……Page 428