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End User License Agreement
by Carlos A. Braumann
Introduction to Stochastic Differential Equations with Applications to Modelling in Biology and Finance
Cover
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
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
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WILEY END USER LICENSE AGREEMENT
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