Abstract
We study a non-linear Hidden Markov Model, where the process of interest is the absolute value of a discretely observed Ornstein–Uhlenbeck diffusion, which is observed after a multiplicative perturbation. We obtain explicit formulae for the recursive relations which link the relevant conditional distributions. As a consequence the predicted, filtered, and smoothed distributions for the hidden process can easily be computed. We illustrate the behaviour of these distributions on simulations.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Cappé O, Moulines E, Rydén T (2005) Inference in Hidden Markov Models. Springer series in statistics. Springer, New York
Chaleyat-Maurel M, Genon-Catalot V (2006) Computable infinite-dimensional filters with applications to discretized diffusion processes. Stoch Process Appl 116(10):1447–1467
Di Masi GB, Runggaldier WJ, Barozzi B (1983) Generalized finite-dimensional filters in discrete time. In: Nonlinear stochastic problems (Algarve, 1982), NATO adv sci inst ser C math phys sci, vol 104. Reidel, Dordrecht, pp 267–277.
Genon-Catalot V (2003) A non-linear explicit filter. Stat Probab Lett 61(2):145–154
Genon-Catalot V, Kessler M (2004) Random scale perturbation of an AR(1) process and its properties as a nonlinear explicit filter. Bernoulli 10(4):701–720
Runggaldier WJ, Spizzichino F (2001) Sufficient conditions for finite dimensionality of filters in discrete time: a Laplace transform-based approach. Bernoulli 7(2):211–221
Sawitzki G (1981) Finite-dimensional filter systems in discrete time. Stochastics 5(1–2):107–114
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Comte, F., Genon-Catalot, V. & Kessler, M. Multiplicative Kalman filtering. TEST 20, 389–411 (2011). https://doi.org/10.1007/s11749-010-0208-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11749-010-0208-0