Introduction to Stochastic Differential Equations with Applications to Modelling in Biology and Finance

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).

• Preface xiAbout the companion website xv1 Introduction 12 Revision of probability and stochastic processes 92.1 Revision of probabilistic concepts 92.2 Monte Carlo simulation of random variables 252.3 Conditional expectations, conditional probabilities, and independence 292.4 A brief review of stochastic processes 352.5 A brief review of stationary processes 402.6 Filtrations, martingales, and Markov times 412.7 Markov processes 453 An informal introduction to stochastic differential equations 514 The Wiener process 574.1 Definition 574.2 Main properties 594.3 Some analytical properties 624.4 First passage times 644.5 Multidimensional Wiener processes 665 Diffusion processes 675.1 Definition 675.2 Kolmogorov equations 695.3 Multidimensional case 736 Stochastic integrals 756.1 Informal definition of the Itô and Stratonovich integrals 756.2 Construction of the Itô integral 796.3 Study of the integral as a function of the upper limit of integration 886.4 Extension of the Itô integral 916.5 Itô theorem and Itô formula 946.6 The calculi of Itô and Stratonovich 1006.7 The multidimensional integral 1047 Stochastic differential equations 1077.1 Existence and uniqueness theorem and main proprieties of the solution 1077.2 Proof of the existence and uniqueness theorem 1117.3 Observations and extensions to the existence and uniqueness theorem 1188 Study of geometric Brownian motion (the stochastic Malthusian model or Black–Scholes model) 1238.1 Study using Itô calculus 1238.2 Study using Stratonovich calculus 1329 The issue of the Itô and Stratonovich calculi 1359.1 Controversy 1359.2 Resolution of the controversy for the particular model 1379.3 Resolution of the controversy for general autonomous models 13910 Study of some functionals 14310.1 Dynkin’s formula 14310.2 Feynman–Kac formula 14611 Introduction to the study of unidimensional Itô diffusions 14911.1 The Ornstein–Uhlenbeck process and the Vasicek model 14911.2 First exit time from an interval 15311.3 Boundary behaviour of Itô diffusions, stationary densities, and first passage times 16012 Some biological and financial applications 16912.1 The Vasicek model and some applications 16912.2 Monte Carlo simulation, estimation and prediction issues 17212.3 Some applications in population dynamics 17912.4 Some applications in fisheries 19212.5 An application in human mortality rates 20113 Girsanov’s theorem 20913.1 Introduction through an example 20913.2 Girsanov’s theorem 21314 Options and the Black–Scholes formula 21914.1 Introduction 21914.2 The Black–Scholes formula and hedging strategy 22614.3 A numerical example and the Greeks 23114.4 The Black–Scholes formula via Girsanov’s theorem 23614.5 Binomial model 24114.6 European put options 24814.7 American options 25114.8 Other models 25315 Synthesis 259References 269Index 277