[eBook Code] Introduction to Stochastic Differential Equations with Applications to Modelling in Biology and Finance (eBook Code, 1st)

Preface xi

About the companion website xv

1 Introduction 1

2 Revision of probability and stochastic processes 9

2.1 Revision of probabilistic concepts 9

2.2 Monte Carlo simulation of random variables 25

2.3 Conditional expectations, conditional probabilities, and independence 29

2.4 A brief review of stochastic processes 35

2.5 A brief review of stationary processes 40

2.6 Filtrations, martingales, and Markov times 41

2.7 Markov processes 45

3 An informal introduction to stochastic differential equations 51

4 The Wiener process 57

4.1 Definition 57

4.2 Main properties 59

4.3 Some analytical properties 62

4.4 First passage times 64

4.5 Multidimensional Wiener processes 66

5 Diffusion processes 67

5.1 Definition 67

5.2 Kolmogorov equations 69

5.3 Multidimensional case 73

6 Stochastic integrals 75

6.1 Informal definition of the Itô and Stratonovich integrals 75

6.2 Construction of the Itô integral 79

6.3 Study of the integral as a function of the upper limit of integration 88

6.4 Extension of the Itô integral 91

6.5 Itô theorem and Itô formula 94

6.6 The calculi of Itô and Stratonovich 100

6.7 The multidimensional integral 104

7 Stochastic differential equations 107

7.1 Existence and uniqueness theorem and main proprieties of the solution 107

7.2 Proof of the existence and uniqueness theorem 111

7.3 Observations and extensions to the existence and uniqueness theorem 118

8 Study of geometric Brownian motion (the stochastic Malthusian model or Black–Scholes model) 123

8.1 Study using Itô calculus 123

8.2 Study using Stratonovich calculus 132

9 The issue of the Itô and Stratonovich calculi 135

9.1 Controversy 135

9.2 Resolution of the controversy for the particular model 137

9.3 Resolution of the controversy for general autonomous models 139

10 Study of some functionals 143

10.1 Dynkin’s formula 143

10.2 Feynman–Kac formula 146

11 Introduction to the study of unidimensional Itô diffusions 149

11.1 The Ornstein–Uhlenbeck process and the Vasicek model 149

11.2 First exit time from an interval 153

11.3 Boundary behaviour of Itô diffusions, stationary densities, and first passage times 160

12 Some biological and financial applications 169

12.1 The Vasicek model and some applications 169

12.2 Monte Carlo simulation, estimation and prediction issues 172

12.3 Some applications in population dynamics 179

12.4 Some applications in fisheries 192

12.5 An application in human mortality rates 201

13 Girsanov’s theorem 209

13.1 Introduction through an example 209

13.2 Girsanov’s theorem 213

14 Options and the Black–Scholes formula 219

14.1 Introduction 219

14.2 The Black–Scholes formula and hedging strategy 226

14.3 A numerical example and the Greeks 231

14.4 The Black–Scholes formula via Girsanov’s theorem 236

14.5 Binomial model 241

14.6 European put options 248

14.7 American options 251

14.8 Other models 253

15 Synthesis 259

References 269

Index 277