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."
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