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In various scientific and industrial fields, stochastic
simulations are taking on a new importance. This is due to the
increasing power of computers and practitioners aim to simulate
more and more complex systems, and thus use random parameters as
well as random noises to model the parametric uncertainties and the
lack of knowledge on the physics of these systems. The error
analysis of these computations is a highly complex mathematical
undertaking. Approaching these issues, the authors present
stochastic numerical methods and prove accurate convergence rate
estimates in terms of their numerical parameters (number of
simulations, time discretization steps). As a result, the book is a
self-contained and rigorous study of the numerical methods within a
theoretical framework. After briefly reviewing the basics, the
authors first introduce fundamental notions in stochastic calculus
and continuous-time martingale theory, then develop the analysis of
pure-jump Markov processes, Poisson processes, and stochastic
differential equations. In particular, they review the essential
properties of Ito integrals and prove fundamental results on the
probabilistic analysis of parabolic partial differential equations.
These results in turn provide the basis for developing stochastic
numerical methods, both from an algorithmic and theoretical point
of view.
The book combines advanced mathematical tools, theoretical analysis
of stochastic numerical methods, and practical issues at a high
level, so as to provide optimal results on the accuracy of Monte
Carlo simulations of stochastic processes. It is intended for
master and Ph.D. students in the field of stochastic processes and
their numerical applications, as well as for physicists,
biologists, economists and other professionals working with
stochastic simulations, who will benefit from the ability to
reliably estimate and control the accuracy of their
simulations."
This volume represents the refereed proceedings of the Sixth
International Conference on Monte Carlo and Quasi-Monte Carlo
Methods in Scienti?c Computing which was held in conjunction with
the Second International C- ference on Monte Carlo and
Probabilistic Methods for Partial Di?erential Equations at
Juan-les-Pins, France, from 7-10 June 2004. The programme of this
conference was arranged by a committee consisting of Henri Faure
(U- versit'
edeMarseille),PaulGlasserman(ColumbiaUniversity),StefanHeinrich
(Universit. at Kaiserslautern), Fred J. Hickernell (Hong Kong
Baptist Univ- sity), Damien Lamberton (Universit' e de Marne la
Vall' ee), Bernard Lapeyre (ENPC-CERMICS), Pierre L'Ecuyer
(Universit'edeMontr' eal), Pierre-Louis Lions (Coll' ege de
France), Harald Niederreiter (National University of S- gapore,
co-chair), Erich Novak (Universit. at Jena), Art B. Owen (Stanford
University), Gilles Pag' es (Universit' e Paris 6), Philip Protter
(Cornell U- versity), Ian H. Sloan (University of New South Wales),
Denis Talay (INRIA Sophia Antipolis, co-chair), Simon Tavar' e
(University of Southern California) and Henryk Wo' zniakowski
(Columbia University and University of Warsaw). The organization of
the conference was arranged by a committee consisting of Mireille
Bossy and Etienne Tanr' e (INRIA Sophia Antipolis), and Madalina
Deaconu(INRIALorraine). LocalarrangementswereinthehandsofMonique
Simonetti and Marie-Line Ramfos (INRIA Sophia Antipolis).
The lecture courses of the CIME Summer School on Probabilistic
Models for Nonlinear PDE's and their Numerical Applications (April
1995) had a three-fold emphasis: first, on the weak convergence of
stochastic integrals; second, on the probabilistic interpretation
and the particle approximation of equations coming from Physics
(conservation laws, Boltzmann-like and Navier-Stokes equations);
third, on the modelling of networks by interacting particle
systems. This book, collecting the notes of these courses, will be
useful to probabilists working on stochastic particle methods and
on the approximation of SPDEs, in particular, to PhD students and
young researchers.
Modeling the Term Structure of Interest Rates provides a
comprehensive review of the continuous-time modeling techniques of
the term structure applicable to value and hedge default-free bonds
and other interest rate derivatives. The authors offer a unifying
framework in which most continuous-time term structure models can
be viewed and compared in terms of their similarities, their
idiosyncratic features, and their main contributions and
limitations. Modeling the Term Structure of Interest Rates is
organized as follows: Section 1 presents the main objectives and
basic definitions and notation; Section 2 proposes an interest rate
models' taxonomy; and Section 3 introduces the mathematical
framework used throughout the survey. Section 4 presents the main
economic theories of the term structure of interest rates. Section
5 provides a unifying framework to present the family of short-term
rate-based term structure models, which encompasses some of the
most popular one and multi-factor short-term rate models developed
at earlier stages of the term structure modeling literature.
Section 6 reviews the second family of forward rate-based models
grouped under the name the Heath, Jarrow and Morton (1992) family
of term structure models. Section 7 presents the recently developed
Libor or market models while preserving the same unifying
mathematical framework. Section 8 surveys the empirical evidence on
interest rate models' estimation and calibration issues and list
selected empirical references which are useful for practitioners
interested in the validity and performance of these models. Section
9 introduces a novel approach to characterize and quantify the
degree of model misspecification associated with interest rate
models. Section 10 discusses some of the challenges posed by
running simulations of continuous-time term structure models.
Section 11 concludes the survey. Finally, some useful mathematical
results can be found in the Appendices.
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