# Introduction to Stochastic Calculus with Applications / Edition 3

This book presents a concise and rigorous treatment of stochastic calculus. It also gives its main applications in finance, biology and engineering. In finance, the stochastic calculus is applied to pricing options by no arbitrage. In biology, it is applied to populations' models, and in engineering it is applied to filter signal from noise. Not everything is

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## Overview

This book presents a concise and rigorous treatment of stochastic calculus. It also gives its main applications in finance, biology and engineering. In finance, the stochastic calculus is applied to pricing options by no arbitrage. In biology, it is applied to populations' models, and in engineering it is applied to filter signal from noise. Not everything is proved, but enough proofs are given to make it a mathematically rigorous exposition.

This book aims to present the theory of stochastic calculus and its applications to an audience which possesses only a basic knowledge of calculus and probability. It may be used as a textbook by graduate and advanced undergraduate students in stochastic processes, financial mathematics and engineering. It is also suitable for researchers to gain working knowledge of the subject. It contains many solved examples and exercises making it suitable for self study.

In the book many of the concepts are introduced through worked-out examples, eventually leading to a complete, rigorous statement of the general result, and either a complete proof, a partial proof or a reference. Using such structure, the text will provide a mathematically literate reader with rapid introduction to the subject and its advanced applications. The book covers models in mathematical finance, biology and engineering. For mathematicians, this book can be used as a first text on stochastic calculus or as a companion to more rigorous texts by a way of examples and exercises.

## Product Details

ISBN-13:
9781848168312
Publisher:
World Scientific Publishing Company, Incorporated
Publication date:
12/01/2011
Pages:
438
Product dimensions:
6.10(w) x 9.00(h) x 1.10(d)

## Related Subjects

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