Appl Math Optim 54:263–264 (2006)
2006 Springer Science+Business Media, Inc.
Stochastic partial differential equations (SPDEs) are coming of age. The initial progress
that essentially deﬁned SPDEs as a ﬁeld was done in the 1970s by a handful of people,
including N. Krylov, E. Pardoux, M. Viot, and J. Walsh. Since then the ﬁeld has grown
substantially. By some estimates, hundreds of mathematicians as well as representatives
of various sciences and engineering are now actively involved in SPDEs.
The development of SPDEs has been heavily inﬂuenced by other ﬁelds of mathemat-
ics (PDEs, interacting particle systems, nonlinear ﬁltering, branching processes) as well
as by sciences and engineering (turbulence, polymers, continuum physics, Euclidean
quantum ﬁeld theory, etc.)
Most of the work done in the 1970s and 1980s had a theoretical ﬂavor, which is quite
understandable. Since the end of the 1980s, papers on numerical methods for SPDEs were
published only sporadically. Mostly, these papers built on the ideas and methodology
already developed for ordinary stochastic differential equations. This trickle of papers
eventually developed in to a larger stream that became a new ﬁeld by itself. A number of
new approaches were introduced that were speciﬁc to SPDEs, and some of them turned
out to be quite efﬁcient. Since SPDEs roots are in applications, the progress in numerical
methods and modeling that we have witnessed lately marks the maturation of the whole
ﬁeld of SPDEs.
This volume is one of the ﬁrst, if not the ﬁrst, attempts to publish a fairly represen-
tative collection of papers focused strictly on numerical methods for SPDEs. It consists
of six papers written by recognized experts in the ﬁeld.
The papers by D. Crisan and S. Lototsky deal with nonlinear ﬁltering (state estima-
tion with noisy observations), which is also known as hidden Markov models (HMM).
This particular ﬁeld has provided a signiﬁcant impetus for the initial development of
SPDEs. It turned out that the posterior distribution of the state given the observations
(the main subject of HMM) is a solution of the so-called Kushner and Zakai SPDEs.
Lototsky and Crisan present two different approaches to computing the posterior
distribution of the state. Lototsky concentrates on the Wiener chaos approach that allows
for the separation of the computations involving the state parameters (known a priori)
from those that require knowledge of the observed quantities. Crisan’s approach is based
on the particle approach to nonlinear ﬁltering. The latter is a dynamical version of the
Markov Chain Monte Carlo.