Introduction to Stochastic Analysis (eBook)

Integrals and Differential Equations

Vigirdas Mackevicius (Autor)

eBook Download: PDF | EPUB

2013 | 1. Auflage
288 Seiten
John Wiley & Sons (Verlag)
978-1-118-60324-6 (ISBN)

Infos

This is an introduction to stochastic integration and stochastic
differential equations written in an understandable way for a wide
audience, from students of mathematics to practitioners in biology,
chemistry, physics, and finances. The presentation is based on the
naïve stochastic integration, rather than on abstract theories
of measure and stochastic processes. The proofs are rather simple
for practitioners and, at the same time, rather rigorous for
mathematicians. Detailed application examples in natural sciences
and finance are presented. Much attention is paid to simulation
diffusion processes.

The topics covered include Brownian motion; motivation of
stochastic models with Brownian motion; Itô and Stratonovich
stochastic integrals, Itô's formula; stochastic
differential equations (SDEs); solutions of SDEs as Markov
processes; application examples in physical sciences and finance;
simulation of solutions of SDEs (strong and weak approximations).
Exercises with hints and/or solutions are also provided.

Autor

Vigirdas Mackevièius is a professor of Faculty of Mathematics and Informatics at Vilnius University in Lithuania.

Inhalt

Preface 9

Notation 13

Chapter 1. Introduction: Basic Notions of Probability Theory17

1.1. Probability space 17

1.2. Random variables 21

1.3. Characteristics of a random variable 21

1.4. Types of random variables 23

1.5. Conditional probabilities and distributions 26

1.6. Conditional expectations as random variables 27

1.7. Independent events and random variables 29

1.8. Convergence of random variables 29

1.9. Cauchy criterion 31

1.10. Series of random variables 31

1.11. Lebesgue theorem 32

1.12. Fubini theorem 32

1.13. Random processes 33

1.14. Kolmogorov theorem 34

Chapter 2. Brownian Motion 35

2.1. Definition and properties 35

2.2. White noise and Brownian motion 45

2.3. Exercises 49

Chapter 3. Stochastic Models with Brownian Motion and WhiteNoise 51

3.1. Discrete time 51

3.2. Continuous time 55

Chapter 4. Stochastic Integral with Respect to Brownian Motion59

4.1. Preliminaries. Stochastic integral with respect to a stepprocess 59

4.2. Definition and properties 69

4.3. Extensions 81

4.4. Exercises 85

Chapter 5. Itô's Formula 87

5.1. Exercises 94

Chapter 6. Stochastic Differential Equations 97

6.1. Exercises 105

Chapter 7. Itô Processes 107

7.1. Exercises 121

Chapter 8. Stratonovich Integral and Equations 125

8.1. Exercises 136

Chapter 9. Linear Stochastic Differential Equations137

9.1. Explicit solution of a linear SDE 137

9.2. Expectation and variance of a solution of an LSDE 141

9.3. Other explicitly solvable equations 145

9.4. Stochastic exponential equation 147

9.5. Exercises 153

Chapter 10. Solutions of SDEs as Markov Diffusion Processes155

10.1. Introduction 155

10.2. Backward and forward Kolmogorov equations 161

10.3. Stationary density of a diffusion process 172

10.4. Exercises 176

Chapter 11. Examples 179

11.1. Additive noise: Langevin equation 180

11.2. Additive noise: general case 180

11.3. Multiplicative noise: general remarks 184

11.4. Multiplicative noise: Verhulst equation 186

11.5. Multiplicative noise: genetic model 189

Chapter 12. Example in Finance: Black-Scholes Model195

12.1. Introduction: what is an option? 195

12.2. Self-financing strategies 197

12.3. Option pricing problem: the Black-Scholes model204

12.4. Black-Scholes formula 206

12.5. Risk-neutral probabilities: alternative derivation ofBlack-Scholes formula 210

12.6. Exercises 214

Chapter 13. Numerical Solution of Stochastic DifferentialEquations 217

13.1. Memories of approximations of ordinary differentialequations 218

13.2. Euler approximation 221

13.3. Higher-order strong approximations 224

13.4. First-order weak approximations 231

13.5. Higher-order weak approximations 238

13.6. Example: Milstein-type approximations 241

13.7. Example: Runge-Kutta approximations 244

13.8. Exercises 249

Chapter 14. Elements of Multidimensional Stochastic Analysis251

14.1. Multidimensional Brownian motion 251

14.2. Itô's formula for a multidimensional Brownianmotion 252

14.3. Stochastic differential equations 253

14.4. Itô processes 254

14.5. Itô's formula for multidimensional Itôprocesses 256

14.6. Linear stochastic differential equations 256

14.7. Diffusion processes 257

14.8. Approximations of stochastic differential equations259

Solutions, Hints, and Answers 261

Bibliography 271

Index 273

Rezensionen

"Thus, the book is a welcome addition in the effort to
make stochastic integration and SDE as accessible as possible to
the greater public interested in or in need of using
them." (Mathematical Reviews, 1 February
2013)

"If I have a chance to teach (again) a course in
stochastic financial modelling, I will definitely choose this to be
among two or three sources to use. I have all the reasons to
strongly recommend it to anybody in the area of modern stochastic
modelling." (Zentralblatt MATH, 1 December 2012)

Produkt-Details

• Erscheint lt. Verlag: 7.2.2013
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Analysis
• Themenwelt: Mathematik / Informatik ► Mathematik ► Wahrscheinlichkeit / Kombinatorik
• Themenwelt: Technik
• Schlagworte: Mathematical Analysis • Mathematics • Mathematik • Mathematische Analyse
• ISBN-10: 1-118-60324-9 / 1118603249
• ISBN-13: 978-1-118-60324-6 / 9781118603246