Carlos A. Braumann - Introduction to Stochastic Differential Equations with Applications to Modelling in Biology and Finance
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- Book:Introduction to Stochastic Differential Equations with Applications to Modelling in Biology and Finance
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Introduction to Stochastic Differential Equations with Applications to Modelling in Biology and Financeoffers 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, It 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 Financeis the authoritative guide to understanding the issues of stochastic differential equations and their application.
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Carlos A. Braumann
University of vora
Portugal
This edition first published 2019
2019 John Wiley & Sons Ltd
All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, except as permitted by law. Advice on how to obtain permission to reuse material from this title is available at http://www.wiley.com/go/permissions.
The right of Carlos A. Braumann to be identified as the author of this work has been asserted in accordance with law.
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Dedication
To Manuela
This is a beginner's book intended as an introduction to stochastic differential equations (SDEs), covering both theory and applications. SDEs are basically differential equations describing the average dynamical behaviour of some phenomenon with an additional stochastic term describing the effect of random perturbations in environmental conditions (environment taken here in a very broad sense) that influence the phenomenon. They have important and increasing applications in basically all fields of science and technology, and they are ubiquitous in modern finance. I feel that the connection between theory and applications is a very powerful tool in mathematical modelling and makes for a better understanding of the theory and its motivations. Therefore, this book illustrates the concepts and theory with several applications. They are mostly reallife applications coming from the biological, bioeconomical, and the financial worlds, based on the research experience (concentrated on biological and bioeconomical applications) and teaching experience of the author and his coworkers, but the methodologies used are of interest to readers interested in applications in other areas and even to readers already acquainted with SDEs.
This book wishes to serve both mathematically strong readers and students, academic community members, and practitioners from different areas (mainly from biology and finance) that wish to use SDEs in modelling. It requires basic knowledge of calculus, ...
This book is accompanied by a companion website:
www.wiley.com/go/braumann/stochasticdifferentialequations
The website includes:
- Solutions to exercises
Scan this QR code to visit the companion website.
Stochastic differential equations (SDEs) are basically differential equations with an additional stochastic term. The deterministic term, which is common to ordinary differential equations, describes the average dynamical behaviour of the phenomenon under study and the stochastic term describes the noise, i.e. the random perturbations that influence the phenomenon. Of course, in the particular case where such random perturbations are absent (deterministic case), the SDE becomes an ordinary differential equation.
As the dynamical behaviour of many natural phenomena can be described by differential equations, SDEs have important applications in basically all fields of science and technology whenever we need to consider random perturbations in the environmental conditions (environment taken here in a very broad sense) that affect such phenomena in a relevant manner.
As far as I know, the first SDE appeared in the literature in Uhlenbeck and Ornstein () explained the phenomenon by the collisions of the particle with the molecules of the fluid and provided a model for the ...
Revision of probability and stochastic processes
Consider a probability space
, where 
is a measurable space and 
is a probability defined on it. Usually, it is a model for a realworld phenomenon or an experiment that depends on chance (i.e. is random) and we shall now see what each element of the triplet 
means.
The universal set or sample space
is a nonempty set containing all possible conditions that may influence the outcome of the random phenomenon or experiment.
If we throw two dice simultaneously, say one white and one black, and are interested in the outcome (number of dots on each of the two dice), the space 
could be the set of all possible physical scenarios describing the throwing of the dice, such as the position of the hands, how strongly and in what direction we throw the dice, the density of the air, and many other factors, some of which we are not even aware. To each such physical scenario there would correspond an outcome in terms of number of dots, but we know little or nothing about the probabilities of the different scenarios or about the correspondence between scenarios and outcomes. Therefore, actually working with this complex space of physical scenarios is not very practical. Fortunately, what really interests us are the actual outcomes determined by the physical scenarios and the probabilities of occurrence of those outcomes. It is therefore legitimate to adopt, as we do, the simplified version of using as 
the much simpler space of the possible outcomes of the throwing of the dice. So, we will use as our sample space the 36element set 
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