526 research outputs found
A Berry-Esseen theorem for Feynman-Kac and interacting particle models
In this paper we investigate the speed of convergence of the fluctuations of
a general class of Feynman-Kac particle approximation models. We design an
original approach based on new Berry-Esseen type estimates for abstract
martingale sequences combined with original exponential concentration estimates
of interacting processes. These results extend the corresponding statements in
the classical theory and apply to a class of branching and genealogical
path-particle models arising in nonlinear filtering literature as well as in
statistical physics and biology.Comment: Published at http://dx.doi.org/10.1214/105051604000000792 in the
Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute
of Mathematical Statistics (http://www.imstat.org
Genealogical particle analysis of rare events
In this paper an original interacting particle system approach is developed
for studying Markov chains in rare event regimes. The proposed particle system
is theoretically studied through a genealogical tree interpretation of
Feynman--Kac path measures. The algorithmic implementation of the particle
system is presented. An estimator for the probability of occurrence of a rare
event is proposed and its variance is computed, which allows to compare and to
optimize different versions of the algorithm. Applications and numerical
implementations are discussed. First, we apply the particle system technique to
a toy model (a Gaussian random walk), which permits to illustrate the
theoretical predictions. Second, we address a physically relevant problem
consisting in the estimation of the outage probability due to polarization-mode
dispersion in optical fibers.Comment: Published at http://dx.doi.org/10.1214/105051605000000566 in the
Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute
of Mathematical Statistics (http://www.imstat.org
Interacting Markov chain Monte Carlo methods for solving nonlinear measure-valued equations
We present a new class of interacting Markov chain Monte Carlo algorithms for
solving numerically discrete-time measure-valued equations. The associated
stochastic processes belong to the class of self-interacting Markov chains. In
contrast to traditional Markov chains, their time evolutions depend on the
occupation measure of their past values. This general methodology allows us to
provide a natural way to sample from a sequence of target probability measures
of increasing complexity. We develop an original theoretical analysis to
analyze the behavior of these iterative algorithms which relies on
measure-valued processes and semigroup techniques. We establish a variety of
convergence results including exponential estimates and a uniform convergence
theorem with respect to the number of target distributions. We also illustrate
these algorithms in the context of Feynman-Kac distribution flows.Comment: Published in at http://dx.doi.org/10.1214/09-AAP628 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Nonasymptotic analysis of adaptive and annealed Feynman-Kac particle models
Sequential and quantum Monte Carlo methods, as well as genetic type search
algorithms can be interpreted as a mean field and interacting particle
approximations of Feynman-Kac models in distribution spaces. The performance of
these population Monte Carlo algorithms is strongly related to the stability
properties of nonlinear Feynman-Kac semigroups. In this paper, we analyze these
models in terms of Dobrushin ergodic coefficients of the reference Markov
transitions and the oscillations of the potential functions. Sufficient
conditions for uniform concentration inequalities w.r.t. time are expressed
explicitly in terms of these two quantities. We provide an original
perturbation analysis that applies to annealed and adaptive Feynman-Kac models,
yielding what seems to be the first results of this kind for these types of
models. Special attention is devoted to the particular case of Boltzmann-Gibbs
measures' sampling. In this context, we design an explicit way of tuning the
number of Markov chain Monte Carlo iterations with temperature schedule. We
also design an alternative interacting particle method based on an adaptive
strategy to define the temperature increments. The theoretical analysis of the
performance of this adaptive model is much more involved as both the potential
functions and the reference Markov transitions now depend on the random
evolution on the particle model. The nonasymptotic analysis of these complex
adaptive models is an open research problem. We initiate this study with the
concentration analysis of a simplified adaptive models based on reference
Markov transitions that coincide with the limiting quantities, as the number of
particles tends to infinity.Comment: Published at http://dx.doi.org/10.3150/14-BEJ680 in the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
Sequential Monte Carlo Methods for Option Pricing
In the following paper we provide a review and development of sequential
Monte Carlo (SMC) methods for option pricing. SMC are a class of Monte
Carlo-based algorithms, that are designed to approximate expectations w.r.t a
sequence of related probability measures. These approaches have been used,
successfully, for a wide class of applications in engineering, statistics,
physics and operations research. SMC methods are highly suited to many option
pricing problems and sensitivity/Greek calculations due to the nature of the
sequential simulation. However, it is seldom the case that such ideas are
explicitly used in the option pricing literature. This article provides an
up-to date review of SMC methods, which are appropriate for option pricing. In
addition, it is illustrated how a number of existing approaches for option
pricing can be enhanced via SMC. Specifically, when pricing the arithmetic
Asian option w.r.t a complex stochastic volatility model, it is shown that SMC
methods provide additional strategies to improve estimation.Comment: 37 Pages, 2 Figure
Sequential Monte Carlo with Highly Informative Observations
We propose sequential Monte Carlo (SMC) methods for sampling the posterior
distribution of state-space models under highly informative observation
regimes, a situation in which standard SMC methods can perform poorly. A
special case is simulating bridges between given initial and final values. The
basic idea is to introduce a schedule of intermediate weighting and resampling
times between observation times, which guide particles towards the final state.
This can always be done for continuous-time models, and may be done for
discrete-time models under sparse observation regimes; our main focus is on
continuous-time diffusion processes. The methods are broadly applicable in that
they support multivariate models with partial observation, do not require
simulation of the backward transition (which is often unavailable), and, where
possible, avoid pointwise evaluation of the forward transition. When simulating
bridges, the last cannot be avoided entirely without concessions, and we
suggest an epsilon-ball approach (reminiscent of Approximate Bayesian
Computation) as a workaround. Compared to the bootstrap particle filter, the
new methods deliver substantially reduced mean squared error in normalising
constant estimates, even after accounting for execution time. The methods are
demonstrated for state estimation with two toy examples, and for parameter
estimation (within a particle marginal Metropolis--Hastings sampler) with three
applied examples in econometrics, epidemiology and marine biogeochemistry.Comment: 25 pages, 11 figure
A Backward Particle Interpretation of Feynman-Kac Formulae
We design a particle interpretation of Feynman-Kac measures on path spaces
based on a backward Markovian representation combined with a traditional mean
field particle interpretation of the flow of their final time marginals. In
contrast to traditional genealogical tree based models, these new particle
algorithms can be used to compute normalized additive functionals "on-the-fly"
as well as their limiting occupation measures with a given precision degree
that does not depend on the final time horizon.
We provide uniform convergence results w.r.t. the time horizon parameter as
well as functional central limit theorems and exponential concentration
estimates. We also illustrate these results in the context of computational
physics and imaginary time Schroedinger type partial differential equations,
with a special interest in the numerical approximation of the invariant measure
associated to -processes
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