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CREST - GENES
Cours de Formation par la Recherche 2008-2009
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Inference for Diffusion Processes Using Monte Carlo
in the Path Space
Omiros PAPASPILIOPOULOS
(Université Pompeu Fabra, Barcelone et Invité CREST, LS)
--------------------Diffusion processes is a large family of time-series models with a wide and increasing range of applications.
They can be used either to model directly observed data, or as a (partially observed or latent) component of
more complex hierarchical models. From a statistical point of view interest lies in the estimation of unknown
parameters of such models and of the process itself when it is only partially observed.
However, inference for partially observed diffusions involves marginal laws (e.g. the transition kernel of
the process) which are typically intractable. This raises serious theoretical and computational challenges.
The course focuses on the computational challenge and develops appropriate Monte Carlo methodology
for parameter and process estimation. The Monte Carlo methods we consider include rejection sampling (RS),
importance sampling (IS) and sequential versions of it, and Markov chain Monte Carlo (MCMC). We show that it
is natural to derive theoretical algorithms in the infinite-dimensional space of the diffusion paths, that is the path
space. Practical implementation of these algorithms can then be achieved either approximately, by finitedimensional projections (discretizations) or by exact retrospective methods.
The infinite-dimensional setup sheds light and gives solutions to problems which are masked in
alternative (and popular) methods which first project to finite-dimensions and then design the Monte Carlo
algorithm. The limiting behaviour (as the approximation gets finer) of such finite-dimensional algorithms often
has serious deficiencies, which include infinite variance of is weights, poor mixing of MCMC algorithms for
simulation of paths, reducibility of MCMC algorithms which update unobserved paths and parameters.
We will also demonstrate how the infinite-dimensional setup justifies certain ad-hoc finite-dimensional
algorithms which have proved successful in this context.
A main aim of the course will be to develop this material and draw connections (some of which have not
been pointed out in the literature) between various works in this line of research.
Bibliographie
A large part of the lecture will be based upon
Papaspiliopoulos, O. et Roberts, G.O. (2008), Monte Carlo Based Inference for Diffusion Processes, Statistics
for Stochastic Differential Equations, Monographs on Statistics and Applied Probability, Chapman and Hall (en
préparation).
Other indicative references include.
Delyon et Hu (2006), « Simulation of Conditioned Diffusion and Application to Parameter Estimation »,
Stochastic Process. Appl. 116, 1660-1675.
Durham et Gallant (2002), « Numerical Techniques for Maximum Likelihood Estimation of Continuous-Time
Diffusion Processes », J. Bus. Econom. Statist. 20, 297-338.
Dacunha-Castelle, D. et D. Florens-Zmirou (1986), « Estimation of the Coefficients of a Diffusion from Discrete
Observations », Stochastic 19, 263-284.
Beskes, A., Papaspiliopoulos, O., Roberts, G.O. et P. Fearnhead (2006), « Exact and Computationally Efficient
Likelihood-Based Estimation for Discretely Observed Diffusion Processes », J.R. Stat. Soc., Ser. B, Stat.
Methodol. 68, 333-382.
Cours Lundi
Les :
Jeudi
Lundi
Jeudi
à l’ENSAE
30
2
6
9
Mars 2009
Avril 2009
Avril 2009
Avril 2009
De 10h à 12h 10
De 9h à 12h 15
De 9h à 12h 15
De 10h à 12h 10
3, Avenue Pierre Larousse, Malakoff
Salle
Salle
Salle
Salle
s8
35
s8
35
(Métro : Malakoff/Plateau de Vanves)
Ces cours sont proposés aux étudiants de 3 ème année de l’ENSAE, de l’ENSAI se préparant à la recherche et ouverts aux étudiants
de M2 ou inscrits en thèse. Une inscription préalable est demandée impérativement pour tous les étudiants de l’ENSAE, de
l’ENSAI, ou extérieurs, par courriel à [email protected] ou par tél. au 01 41 17 35 50, afin de pouvoir être admis dans les locaux
de l’ENSAE. Les renseignements supplémentaires sur le contenu et les dates de ces cours peuvent être obtenus au 0141173550.
CREST-OFPR, Bureau 2107, 15 Boulevard Gabriel Péri, 92245 MALAKOFF Cédex.