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More About This Title Introduction to Stochastic Differential Equationswith Applications to Modelling in Biology andFinance
English
A comprehensive introduction to the core issues of stochastic differential equations and their effective application
Introduction to Stochastic Differential Equations with Applications to Modelling in Biology and Finance offers a comprehensive examination to the most important issues of stochastic differential equations and their applications. The author — a noted expert in the field — includes myriad illustrative examples in modelling dynamical phenomena subject to randomness, mainly in biology, bioeconomics and finance, that clearly demonstrate the usefulness of stochastic differential equations in these and many other areas of science and technology.
The text also features real-life situations with experimental data, thus covering topics such as Monte Carlo simulation and statistical issues of estimation, model choice and prediction. The book includes the basic theory of option pricing and its effective application using real-life. The important issue of which stochastic calculus, Itô or Stratonovich, should be used in applications is dealt with and the associated controversy resolved. Written to be accessible for both mathematically advanced readers and those with a basic understanding, the text offers a wealth of exercises and examples of application. This important volume:
- Contains a complete introduction to the basic issues of stochastic differential equations and their effective application
- Includes many examples in modelling, mainly from the biology and finance fields
- Shows how to: Translate the physical dynamical phenomenon to mathematical models and back, apply with real data, use the models to study different scenarios and understand the effect of human interventions
- Conveys the intuition behind the theoretical concepts
- Presents exercises that are designed to enhance understanding
- Offers a supporting website that features solutions to exercises and R code for algorithm implementation
Written for use by graduate students, from the areas of application or from mathematics and statistics, as well as academics and professionals wishing to study or to apply these models, Introduction to Stochastic Differential Equations with Applications to Modelling in Biology and Finance is the authoritative guide to understanding the issues of stochastic differential equations and their application.
English
CARLOS A. BRAUMANN is Professor in the Department of Mathematics and member of the Research Centre in Mathematics and Applications, Universidade de Évora, Portugal. He is an elected member of the International Statistical Institute (since 1992), a former President of the European Society for Mathematical and Theoretical Biology (2009-12) and of the Portuguese Statistical Society (2006-09 and 2009-12), and a former member of the European Regional Committee of the Bernoulli Society (2008-12). He has dealt with stochastic differential equation (SDE) models and applications (mainly biological).
English
1 Introduction 1
2 Revision of probability and stochastic processes 7
2.1 Revision of probabilistic concepts 7
2.2 Monte Carlo simulation of random variables 21
2.3 Conditional expectations and probabilities. Independence 26
2.4 A brief review of stochastic processes 32
2.5 A brief review of stationary processes 37
2.6 Filtrations, martingales and Markov times 38
2.7 Markov processes 42
3 An informal introduction to stochastic differential equations 49
4 The Wiener process 55
4.1 Definition 55
4.2 Main properties 57
4.3 Some analytical properties 60
4.4 First passage times 61
4.5 Multidimensional Wiener processes 63
5 Diffusion processes 65
5.1 Definition 65
5.2 Kolmogorov equations 66
5.3 Multidimensional case 71
6 Stochastic integrals 73
6.1 Informal definition of the Itô and the Stratonovich integrals 73
6.2 Construction of the Itô integral 77
6.3 Study of the integral as a function of the upper limit of integration 86
6.4 Extension of the Itô integral 88
6.5 Itô theorem amd Itô formula 91
6.6 The calculi of Itô and Stratonovich 96
6.7 The multidimensional integral 100
7 Stochastic differential equations 103
7.1 Existence and uniqueness theorem and main proprieties of the solution 103
7.2 Proof of the existence and uniqueness theorem 107
7.3 Observations and extensions to the existence and uniqueness theorem 113
8 Study of geometric Brownian motion (the stochastic Malthusian model or Black-Scholes model) 119
8.1 Study using Itô calculus 119
8.2 Study using Stratonovich calculus 127
9 The issue of the Itô and Stratonovich calculi 129
9.1 Controversy 129
9.2 Resolution of the controversy for the particular model 131
9.3 Resolution of the controversy for general autonomous models 133
10 Study of some functionals 137
10.1 Dynkin’s formula 137
10.2 Feynman-Kac formula 140
11 Introduction to the study of unidimensional Itô diffusions 143
11.1 The Ornstein-Uhlenbeck process and the Vasicek model 143
11.2 First exit time from an interval 146
11.3 Boundary behavior of Itô diffusions, stationary densities and first passage times 153
12 Some biological and financial applications 161
12.1 Vasicek model and some applications 161
12.2 Monte Carlo simulation, estimation and prediction issues 163
12.3 Some applications in population dynamics 170
12.4 Some applications in fisheries 182
12.5 An application in human mortality rates 191
13 Girsanov’s theorem 197
13.1 Introduction through an example 197
13.2 Girsanov’s theorem 201
14 Options and Black-Scholes formula 207
14.1 Introduction 207
14.2 Black-Scholes formula and hedging strategy 213
14.3 A numerical example and the Greeks 218
14.4 Black-Scholes formula via Girsanov’s theorem 223
14.5 Binomial model 228
14.6 European put options 235
14.7 American options 237
14.8 Other models 240
15 Synthesis 245
Index 261
