Abstract. In this paper the filtering of partially observed diffusions, with discrete-time observations, is considered. It is assumed that only biased approximations of the diffusion can be obtained, for choice of an accuracy parameter indexed by l. A multilevel estimator is proposed, consisting of a telescopic sum of increment estimators associated to the successive levels. The work associated to O(ε 2 ) mean-square error between the multilevel estimator and average with respect to the filtering distribution is shown to scale optimally, for example as O(ε −2 ) for optimal rates of convergence of the underlying diffusion approximation. The method is illustrated on some toy examples as well as estimation of interest rate based on real S&P 500 stock price data.
Beskos et al. AbstractWe consider the numerical approximation of the filtering problem in high dimensions, that is when the hidden state lies in R d with d large. For low dimensional problems, one of the most popular numerical procedures for consistent inference is the class of approximations termed particle filters or sequential Monte Carlo methods. However, in high dimensions, standard particle filters (e.g. the bootstrap particle filter) can have a cost that is exponential in d for the algorithm to be stable in an appropriate sense. We develop a new particle filter, called the space-time particle filter, for a specific family of state-space models in discrete time. This new class of particle filters provide consistent Monte Carlo estimates for any fixed d, as do standard particle filters. Moreover, when there is a spatial mixing element in the dimension of the state-vector, the space-time particle filter will scale much better with d than the standard filter for a class of filtering problems. We illustrate this analytically for a model of a simple i.i.d. structure and one ofspace-direction, when we show that the algorithm exhibits certain stability properties as d increases at a cost O(nN d 2 ), where n is the time parameter and N is the number of Monte Carlo samples, that are fixed and independent of d. Our theoretical results are also supported by numerical simulations on practical models of complex structures. The results suggest that it is indeed possible to tackle some high dimensional filtering problems using the space-time particle filter that standard particle filters cannot handle.
This article is concerned with the fluctuation analysis and the stability properties of a class of one-dimensional Riccati diffusions. These one-dimensional stochastic differential equations exhibit a quadratic drift function and a non-Lipschitz continuous diffusion function. We present a novel approach, combining tangent process techniques, Feynman-Kac path integration, and exponential change of measures, to derive sharp exponential decays to equilibrium. We also provide uniform estimates with respect to the time horizon, quantifying with some precision the fluctuations of these diffusions around a limiting deterministic Riccati differential equation. These results provide a stronger and almost sure version of the conventional central limit theorem. We illustrate these results in the context of ensemble Kalman-Bucy filtering. To the best of our knowledge, the exponential stability and the fluctuation analysis developed in this work are the first results of this kind for this class of nonlinear diffusions.
In this article we consider computing expectations w.r.t. probability laws associated to a certain class of stochastic systems. In order to achieve such a task, one must not only resort to numerical approximation of the expectation, but also to a biased discretization of the associated probability. We are concerned with the situation for which the discretization is required in multiple dimensions, for instance in space-time.In such contexts, it is known that the multi-index Monte Carlo (MIMC) method of [7] can improve upon i.i.d. sampling from the most accurate approximation of the probability law. Through a non-trivial modification of the multilevel Monte Carlo (MLMC) method, this method can reduce the work to obtain a given level of error, relative to i.i.d. sampling and relative even to MLMC. In this article we consider the case when such probability laws are too complex to be sampled independently, for example a Bayesian inverse problem where evaluation of the likelihood requires solution of a partial differential equation (PDE) model which needs to be approximated at finite resolution. We develop a modification of the MIMC method which allows one to use standard Markov chain Monte Carlo (MCMC) algorithms to replace independent and coupled sampling, in certain contexts. We prove a variance theorem for a simplified estimator which shows that using our MIMCMC method is preferable, in the sense above, to i.i.d. sampling from the most accurate approximation, under appropriate assumptions. The method is numerically illustrated on a Bayesian inverse problem associated to a stochastic partial differential equation (SPDE), where the path measure is conditioned on some observations.
In this article we introduce two new estimates of the normalizing constant (or marginal likelihood) for partially observed diffusion (POD) processes, with discrete observations. One estimate is biased but non-negative and the other is unbiased but not almost surely non-negative. Our method uses the multilevel particle filter of [11].We show that, under assumptions, for Euler discretized PODs and a given ε > 0 in order to obtain a mean square error (MSE) of O(ε 2 ) one requires a work of O(ε −2.5 ) for our new estimates versus a standard particle filter that requires a work of O(ε −3 ).Our theoretical results are supported by numerical simulations.
In this article we consider static Bayesian parameter estimation for partially observed diffusions that are discretely observed. We work under the assumption that one must resort to discretizing the underlying diffusion process, for instance using the Euler Maruyama method.Given this assumption, we show how one can use Markov chain Monte Carlo (MCMC) and particularly particle MCMC [Andrieu, C., Doucet, A. & Holenstein, R. (2010). Particle Markov chain Monte Carlo methods (with discussion). J. R. Statist. Soc. Ser. B, 72, 269-342] to implement a new approximation of the multilevel (ML) Monte Carlo (MC) collapsingsum identity. Our approach comprises constructing an approximate coupling of the posterior density of the joint distribution over parameter and hidden variables at two different discretization levels and then correcting by an importance sampling method. The variance of the weights are independent of the length of the observed data set. The utility of such a method is that, for a prescribed level of mean square error, the cost of this MLMC is provably less than i.i.d. sampling from the posterior associated to the most precise discretization. However the method here comprises using only known and efficient simulation methodologies. The theoretical results are illustrated by inference of the parameters of two prototypical processes given noisy partial observations of the process: the first is an Ornstein Uhlenbeck process and the second is a more general Langevin equation.
The YUIMA Project is an open source and collaborative effort aimed at developing the R package yuima for simulation and inference of stochastic differential equations. In the yuima package stochastic differential equations can be of very abstract type, multidimensional, driven by Wiener process or fractional Brownian motion with general Hurst parameter, with or without jumps specified as Lévy noise. The yuima package is intended to offer the basic infrastructure on which complex models and inference procedures can be built on. This paper explains the design of the yuima package and provides some examples of applications.
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