Stochastic Differential Equations: Theory and Practic of Numerical Solution. With MATLAB Programs
Author(s):
Dmitriy Feliksovich Kuznetsov
Peter the Great Saint-Petersburg Polytechnic University
Russia, 195251, Saint-Petersburg, Polytechnicheskaya st., 29
Department of Higher Mathematics
Professor, Doctor of Physico-Mathematical Sciences
sde_kuznetsov@inbox.ru
Abstract:
This is the sixth (revised and expanded) edition of the book
"Stochastic differential equations: theory and practice of numerical solution".
The monograph is devoted to the problem of the numerical integration
of stochastic differential equations (SDE). The case of the Ito
SDE is systematically analised and the case of SDE with jump component
is considered as well. In the book the effective approach to the
numerical integration of the Ito SDE, which is based on the Taylor-Ito and
the Taylor-Stratonovich expansions,
is systematically analised.
One of the aims of this monograph is the
development and application of the Fourier method to the numerical
solution of the Ito SDE. The Fourier series are widely used in various fields of
applied mathematics and physics. However, the method of the Fourier series
as applied to the numerical solution of the Ito SDE has not been adequately
studied.
This monograph partially fills this gap. The book consists of 17 chapters.
In chapter 1 we present additional material which may be used
while reading this book. In chapter 2 the mathematical models
of dynamical systems of different physical nature under the influence of
random disturbances on the base of SDE are considered.
Some mathematical problems for SDE are described as well.
Chapter 3 is devoted to some properties and formulas
for stochastic integrals. In chapter 4 we discuss the stochastic
Taylor expansions.
We consider the classical Taylor-Ito and Taylor-Stratonovich
expansions and 4 new, so called unified the Taylor-Ito and the Taylor-Stratonovich
expansions.
Chapters 5 and 6 are devoted to strong (mean-square) approximations
of collections of the iterative Ito and Stratonovich stochastic
integrals from the Taylor-Ito and Taylor-Stratonovich expansions.
In chapters 7-9 we construct strong numerical methods, and
in chapter 10 we consider weak numerical methods for the Ito SDE.
Chapter 11 is devoted to the numerical integration of linear stationary
systems of the Ito SDE. In chapter 12 we consider the theory of numerical
integration of SDE with jump component. In chapter 13 we provide the library
of MATLAB programs for the numerical integration of linear stationary systems
of the Ito SDE.
In chapters 14-16 the application of the numerical methods constructed
in the monograph to the modeling selected trajectories
of solutions of the non-linear Ito SDE systems (chapter 14), to numerical
solution of mathematical problems by strong (chapter 15) and weak
(chapter 16) numerical methods is demonstrated.
For the first time the numerical modeling
the iterative Ito and Stratonovich stochastic integrals is realised
with the usage of the Legendre polynomial system.
Chapter 17 contains full texts of MATLAB programs,
implementing the numerical experiments along the text of the whole book.
Keywords
- explicit numerical method
- finite-difference numerical method
- implicit numerical method
- Ito stochastic differential equation
- Ito stochastic integral
- Legendre polynomial
- MATLAB program
- mean-square convergence
- multiple Fourier series
- multiple Fourier-Legemdre series
- multiple Ito stochastic integral
- multiple stochastic integral expansion
- multiple Stratonovich stochastic integral
- multiple trigonometric Fourier series
- numerical integration
- numerical method
- numerical method of Runge-Kutta type
- numerical modeling
- one-step numerical method
- Parseval equality
- stochastic differential equation
- stochastic differential equation with jump component
- stochastic integral on martingale
- stochastic integral on Poisson measure
- stochastic Taylor expansion
- Stratonovich stochastic integral
- strong approximation
- strong convergence
- strong numerical method
- Taylor-Ito expansion
- Taylor-Stratonovich expansion
- three-step numerical method
- two-step numerical method
- unified Taylor-Ito expansion
- unified Taylor-Stratonovich expansion
- weak approximation
- weak convergence
- weak numerical method
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