Introduction to stochastic differential equations with applications to modelling in biology and finance
A comprehensive introduction to the core issues of stochastic differential equations and their effective applicationIntroduction to Stochastic Differential Equations with Applications to Modelling in Biology and Finance offers a comprehensive examination to the most important issues of stochastic di...
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| Format: | E-Book Buch |
| Sprache: | Englisch |
| Veröffentlicht: |
Hoboken, NJ
Wiley
2019
John Wiley & Sons, Incorporated Wiley-Blackwell |
| Ausgabe: | 1 |
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| ISBN: | 1119166063, 9781119166061 |
| Online-Zugang: | Volltext |
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Inhaltsangabe:
- 11 Introduction to the study of unidimensional Itô diffusions -- 11.1 The Ornstein-Uhlenbeck process and the Vasicek model -- 11.2 First exit time from an interval -- 11.3 Boundary behaviour of Itô diffusions, stationary densities, and first passage times -- 12 Some biological and financial applications -- 12.1 The Vasicek model and some applications -- 12.2 Monte Carlo simulation, estimation and prediction issues -- 12.3 Some applications in population dynamics -- 12.4 Some applications in fisheries -- 12.5 An application in human mortality rates -- 13 Girsanov's theorem -- 13.1 Introduction through an example -- 13.2 Girsanov's theorem -- 14 Options and the Black-Scholes formula -- 14.1 Introduction -- 14.2 The Black-Scholes formula and hedging strategy -- 14.3 A numerical example and the Greeks -- 14.4 The Black-Scholes formula via Girsanov's theorem -- 14.5 Binomial model -- 14.6 European put options -- 14.7 American options -- 14.8 Other models -- 15 Synthesis -- References -- Index -- End User License Agreement
- Intro -- Table of Contents -- Preface -- About the companion website -- 1 Introduction -- 2 Revision of probability and stochastic processes -- 2.1 Revision of probabilistic concepts -- 2.2 Monte Carlo simulation of random variables -- 2.3 Conditional expectations, conditional probabilities, and independence -- 2.4 A brief review of stochastic processes -- 2.5 A brief review of stationary processes -- 2.6 Filtrations, martingales, and Markov times -- 2.7 Markov processes -- 3 An informal introduction to stochastic differential equations -- 4 The Wiener process -- 4.1 Definition -- 4.2 Main properties -- 4.3 Some analytical properties -- 4.4 First passage times -- 4.5 Multidimensional Wiener processes -- 5 Diffusion processes -- 5.1 Definition -- 5.2 Kolmogorov equations -- 5.3 Multidimensional case -- 6 Stochastic integrals -- 6.1 Informal definition of the Itô and Stratonovich integrals -- 6.2 Construction of the Itô integral -- 6.3 Study of the integral as a function of the upper limit of integration -- 6.4 Extension of the Itô integral -- 6.5 Itô theorem and Itô formula -- 6.6 The calculi of Itô and Stratonovich -- 6.7 The multidimensional integral -- 7 Stochastic differential equations -- 7.1 Existence and uniqueness theorem and main proprieties of the solution -- 7.2 Proof of the existence and uniqueness theorem -- 7.3 Observations and extensions to the existence and uniqueness theorem -- 8 Study of geometric Brownian motion (the stochastic Malthusian model or Black-Scholes model) -- 8.1 Study using Itô calculus -- 8.2 Study using Stratonovich calculus -- 9 The issue of the Itô and Stratonovich calculi -- 9.1 Controversy -- 9.2 Resolution of the controversy for the particular model -- 9.3 Resolution of the controversy for general autonomous models -- 10 Study of some functionals -- 10.1 Dynkin's formula -- 10.2 Feynman-Kac formula