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

Author Carlos A. Braumann
Release Date: 2019/04/01
ISBN: 9781119166061
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Book Description

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.

Table of Contents

  1. Cover
  2. Preface
  3. About the companion website
  4. 1 Introduction
  5. 2 Revision of probability and stochastic processes
    1. 2.1 Revision of probabilistic concepts
    2. 2.2 Monte Carlo simulation of random variables
    3. 2.3 Conditional expectations, conditional probabilities, and independence
    4. 2.4 A brief review of stochastic processes
    5. 2.5 A brief review of stationary processes
    6. 2.6 Filtrations, martingales, and Markov times
    7. 2.7 Markov processes
  6. 3 An informal introduction to stochastic differential equations
  7. 4 The Wiener process
    1. 4.1 Definition
    2. 4.2 Main properties
    3. 4.3 Some analytical properties
    4. 4.4 First passage times
    5. 4.5 Multidimensional Wiener processes
  8. 5 Diffusion processes
    1. 5.1 Definition
    2. 5.2 Kolmogorov equations
    3. 5.3 Multidimensional case
  9. 6 Stochastic integrals
    1. 6.1 Informal definition of the Itô and Stratonovich integrals
    2. 6.2 Construction of the Itô integral
    3. 6.3 Study of the integral as a function of the upper limit of integration
    4. 6.4 Extension of the Itô integral
    5. 6.5 Itô theorem and Itô formula
    6. 6.6 The calculi of Itô and Stratonovich
    7. 6.7 The multidimensional integral
  10. 7 Stochastic differential equations
    1. 7.1 Existence and uniqueness theorem and main proprieties of the solution
    2. 7.2 Proof of the existence and uniqueness theorem
    3. 7.3 Observations and extensions to the existence and uniqueness theorem
  11. 8 Study of geometric Brownian motion (the stochastic Malthusian model or Black–Scholes model)
    1. 8.1 Study using Itô calculus
    2. 8.2 Study using Stratonovich calculus
  12. 9 The issue of the Itô and Stratonovich calculi
    1. 9.1 Controversy
    2. 9.2 Resolution of the controversy for the particular model
    3. 9.3 Resolution of the controversy for general autonomous models
  13. 10 Study of some functionals
    1. 10.1 Dynkin's formula
    2. 10.2 Feynman–Kac formula
  14. 11 Introduction to the study of unidimensional Itô diffusions
    1. 11.1 The Ornstein–Uhlenbeck process and the Vasicek model
    2. 11.2 First exit time from an interval
    3. 11.3 Boundary behaviour of Itô diffusions, stationary densities, and first passage times
  15. 12 Some biological and financial applications
    1. 12.1 The Vasicek model and some applications
    2. 12.2 Monte Carlo simulation, estimation and prediction issues
    3. 12.3 Some applications in population dynamics
    4. 12.4 Some applications in fisheries
    5. 12.5 An application in human mortality rates
  16. 13 Girsanov's theorem
    1. 13.1 Introduction through an example
    2. 13.2 Girsanov's theorem
  17. 14 Options and the Black–Scholes formula
    1. 14.1 Introduction
    2. 14.2 The Black–Scholes formula and hedging strategy
    3. 14.3 A numerical example and the Greeks
    4. 14.4 The Black–Scholes formula via Girsanov's theorem
    5. 14.5 Binomial model
    6. 14.6 European put options
    7. 14.7 American options
    8. 14.8 Other models
  18. 15 Synthesis
  19. References
  20. Index
  21. End User License Agreement
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