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One service mathematics has rendered the 'Et moi, ... si j'avait su
comment en revenir, human race. It has put common sense back je n'y
serais point aIle.' Jules Verne where it belongs, on the topmost
shelf next to the dusty canister labelled 'discarded non sense'.
The series is divergent; thererore we may be able to do something
with it. Eric T. Bell O. Heaviside Mathematics is a tool for
thought. A highly necessary tool in a world where both feedback and
non Iinearities abound. Similarly, all kinds of parts of
mathematics serve as tools for other parts and for other sciences.
Applying a simple rewriting rule to the quote on the right above
one finds such statements as: 'One service topology has rendered
mathematical physics .. .'; 'One service logic has rendered com
puter science .. .'; 'One service category theory has rendered
mathematics .. .'. All arguably true. And all statements obtainable
this way form part of the raison d'etre of this series."
This volume deals with the analysis of nonlinear evolution problems
described by partial differential equations having random or
stochastic parameters. The emphasis throughout is on the actual
determination of solutions, rather than on proving the existence of
solutions, although mathematical proofs are given when this is
necessary from an applications point of view. The content is
divided into six chapters. Chapter 1 gives a general presentation
of mathematical models in continuum mechanics and a description of
the way in which problems are formulated. Chapter 2 deals with the
problem of the evolution of an unconstrained system having random
space-dependent initial conditions, but which is governed by a
deterministic evolution equation. Chapter 3 deals with the
initial-boundary value problem for equations with random initial
and boundary conditions as well as with random parameters where the
randomness is modelled by stochastic separable processes. Chapter 4
is devoted to the initial-boundary value problem for models with
additional noise, which obey Ito-type partial differential
equations. Chapter 5 is essential devoted to the qualitative and
quantitative analysis of the chaotic behaviour of systems in
continuum physics. Chapter 6 provides indications on the solution
of ill-posed and inverse problems of stochastic type and suggests
guidelines for future research. The volume concludes with an
Appendix which gives a brief presentation of the theory of
stochastic processes. Examples, applications and case studies are
given throughout the book and range from those involving simple
stochasticity to stochastic illposed problems. For applied
mathematicians, engineers and physicists whosework involves solving
stochastic problems.
This book is a final year undergraduate text on stochastic processes, a tool used widely by statisticians and researchers working in the mathematics of finance. The book will give a detailed treatment of conditional expectation and probability, a topic which in principle belongs to probability theory, but is essential as a tool for stochastic processes. Although the book is a final year text, the author has chosen to use exercises as the main means of explanation for the various topics, and the book will have a strong self-study element. The author has concentrated on the major topics within stochastic analysis: Stochastic Processes, Markov Chains, Spectral Theory, Renewal Theory, Martingales and Itô Stochastic Processes.
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