ESAIM: PROCEEDINGS, September 2007, Vol.19, 108-114 Christophe Andrieu & Dan Crisan, Editors DOI: 10.1051/proc:071914 PARTICLE FILTERS FOR MULTISCALE DIFFUSIONS ∗, ∗∗ Anastasia Papavasiliou 1 Abstract. We consider multiscale stochastic systems that are partially observed at discrete points of the slow time scale. We introduce a particle filter that takes advantage of the multiscale structure of the system to efficiently approximate the optimal filter. R´ esum´ e. On consid` ere des syst` emes stochastiques multi-´ echelles qui sont partiellement observ´ es`a des points discrets de l’´ echelle de temps lente. Onintroduit unfiltre `a particule qui utilise la structure multi-´ echelle du syst` eme pour approcher efficacement le filtre optimal. Introduction We are interested in the problem of estimating a function of a multiscale process that can be approximated by a diffusion which lives the slow scale, when it is partially observed. Such problems come up in many applications, such as molecular dynamics, climate modelling or estimation of stochastic volatility using agent-based models (see [8] for general discussion of multiscale models or [4] and [9] for applications to kinetic Monte-Carlo and climate modelling respectively). In this paper, we focus on the problem of estimating the slow component of a continuous multiscale process from partial and discrete observations of it. More specifically, we have an R p+q process X ǫ =(X ǫ t ) t≥0 = (X (1,ǫ) t ,X (2,ǫ) t ) t≥0 that satisfies the following multiscale stochastic differential equation:  dX (1,ǫ) t = a(X (1,ǫ) t ,X (2,ǫ) t )dt + σ 1 (X (1,ǫ) t ,X (2,ǫ) t )dW (1) t dX (2,ǫ) t = 1 ǫ b(X (1,ǫ) t ,X (2,ǫ) t )dt + 1 √ ǫ σ 2 (X (1,ǫ) t ,X (2,ǫ) t )dW (2) t (1) where X (1,ǫ) t ∈ R p , X (2,ǫ) t ∈ R q and W (1) t and W (2) t are two independent Wiener processes in R p and R q respectively. Let μ be the initial distribution, i.e. μ = L(X ǫ 0 ). We denote by μ 1 and μ 2 the marginals on X (1,ǫ) 0 and X (2,ǫ) 0 respectively. We observe the process (X ǫ t ) t≥0 through Y ǫ =(Y ǫ kΔ ) k=0,...,T , where ∆ ∼O(1), i.e. the observations live in the the same scale as X (1,ǫ) t , which we call the slow time scale. In fact, let us assume for simplicity that ∆ = 1. The process Y ǫ is given by Y ǫ k = h(X (1,ǫ) k ,v k ), (2) where (v k ) k are i.i.d. random variables with known distribution. * The author has been partially supported by a Marie Curie International Reintegration Grant, MIRG-CT-2005-029160. ** The author would like to thank Professor I.G. Kevrekidis for suggesting this problem to her. 1 Department of Statistics, University of Warwick, Coventry, CV4 7AL, UK. Email: a.papavasiliou@warwick.ac.uk c  EDP Sciences, SMAI 2007 Article published by EDP Sciences and available at http://www.edpsciences.org/proc or http://dx.doi.org/10.1051/proc:071914