Introduction to stochastic differential equations with applications to modelling in biology and finance /
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| Author / Creator: | Braumann, Carlos A., 1951- author. |
|---|---|
| Imprint: | Hoboken, NJ : John Wiley & Sons, Inc., 2019. ©2019 |
| Description: | 1 online resource |
| Language: | English |
| Subject: | |
| Format: | E-Resource Book |
| URL for this record: | http://pi.lib.uchicago.edu/1001/cat/bib/13664601 |
Table of Contents:
- 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 properties4.4 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 solution7.2 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 formula10.2 Feynman-Kac formula; 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 theorem13.1 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