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# Multilevel particle filters for the non-linear filtering problem in continuous time

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## Abstract and Figures

In the following article we consider the numerical approximation of the non-linear filter in continuous-time, where the observations and signal follow diffusion processes. Given access to high-frequency, but discrete-time observations, we resort to a first order time discretization of the non-linear filter, followed by an Euler discretization of the signal dynamics. In order to approximate the associated discretized non-linear filter, one can use a particle filter. Under assumptions, this can achieve a mean square error of $$\mathcal {O}(\epsilon ^2)$$, for $$\epsilon >0$$ arbitrary, such that the associated cost is $$\mathcal {O}(\epsilon ^{-4})$$. We prove, under assumptions, that the multilevel particle filter of Jasra et al. (SIAM J Numer Anal 55:3068–3096, 2017) can achieve a mean square error of $$\mathcal {O}(\epsilon ^2)$$, for cost $$\mathcal {O}(\epsilon ^{-3})$$. This is supported by numerical simulations in several examples.
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Statistics and Computing (2020) 30:1381–1402
https://doi.org/10.1007/s11222-020-09951-9
Multilevel particle ﬁlters for the non-linear ﬁltering problem in
continuous time
Ajay Jasra1·Fangyuan Yu1·Jeremy Heng2
Received: 15 July 2019 / Accepted: 27 May 2020 / Published online: 15 June 2020
Abstract
In the following article we consider the numerical approximation of the non-linear ﬁlter in continuous-time, where the
to a ﬁrst order time discretization of the non-linear ﬁlter, followed by an Euler discretization of the signal dynamics. In order
to approximate the associated discretized non-linear ﬁlter, one can use a particle ﬁlter. Under assumptions, this can achieve a
mean square error of O(2),for>0 arbitrary, such that the associated cost is O(4). We prove, under assumptions, that
the multilevel particle ﬁlter of Jasra et al. (SIAM J Numer Anal 55:3068–3096, 2017) can achieve a mean square error of
O(2), for cost O( 3). This is supported by numerical simulations in several examples.
Keywords Multilevel Monte Carlo ·Particle ﬁlters ·Non-linear ﬁltering
1 Introduction
The non-linear ﬁltering problem in continuous-time is found
in many applications in ﬁnance, economics and engineering;
see e.g. Bain and Crisan (2009). We consider the case where
one seeks to ﬁlter an unobserved diffusion process (the sig-
continuous in time and following a diffusion process itself.
The non-linear ﬁlter is the solution to the Kallianpur–Striebel
formula (e.g. Bain and Crisan 2009) and typically has no ana-
lytical solution. This has led to a substantial literature on the
numerical solution of the ﬁltering problem; see for instance
(Bain and Crisan 2009;DelMoral2013).
vations, but not an entire trajectory and this often means one
has to time discretize the functionals associated to the path of
BJeremy Heng
b00760223@essec.edu
Ajay Jasra
ajay.jasra@kaust.edu.sa
Fangyuan Yu
fangyuan.yu@kaust.edu.sa
1Computer, Electrical and Mathematical Sciences and
Engineering Division, King Abdullah University of Science
and Technology, Thuwal 23955, Kingdom of Saudi Arabia
2ESSEC Business School, Singapore 139408, Singapore
the observation and signal. This latter task can be achieved
by using the approach in Picard (1984), which is the one
(Crisan and Ortiz-Latorre 2013,2019). Even under such a
time-discretization, such a ﬁlter is not available analytically,
for most problems of practical interest. From here one must
often discretize the dynamics of the signal (such as Euler),
which in essence leads to a high-frequency discrete-time non-
linear ﬁlter. This latter object can be approximated using
particle ﬁlters in discrete time, as in, for instance, Bain and
Alternatives exist, such as unbiased methods Fearnhead et al.
(2010) and integration-by-parts, change of variables along
with Feynman–Kac particle methods Del Moral (2013), but,
each of these schemes has its advantages and pitfalls versus
the one followed in this paper. We refer to e.g. Crisan and
Ortiz-Latorre (2019) for some discussion.
Particle ﬁlters generate Nsamples (or particles) in par-
allel and sequentially approximate non-linear ﬁlters using
sampling and resampling. The algorithms are very well
understood mathematically; see for instance Del Moral
(2013) and the references therein. Given the particle ﬁlter
approximation of the time-discretized ﬁlter, using an Euler
method for the signal, one can expect that to obtain a mean
squared error (MSE), relative to the true ﬁlter, of O(2),for
>0 arbitrary, the associated cost is O(4). This follows
from standard results on discretizations and particle ﬁlters.
123
... The approach improves performance and is robust to arbitrarily small time-discretization at the expense of additional computational cost. The latter is reduced using a new Multilevel Particle Filter; see [17,20] for some existing approaches. This article is structured as follows. ...
... One could focus on a time-discretization of either side of (2.5), however, as is conventional in the literature (e.g. [1,20]) we focus on the left hand side. ...
... We note that to choose l as specified, one has to have access to an appropriately finely observed data set and this is assumed throughout. Typically, one could use a multilevel Monte Carlo method, as in [20], to reduce the cost to achieve an MSE of O( 2 ). However, in this case as the O(N 2 ) cost dominates and does not depend on l, one can easily check that such a variance reduction method will not improve our particle method. ...
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... Another common recognised problem in continuous time filtering for diffusion processes is the unavailability of transition densities [14,19]. In our problem though, the hidden state is described by a linear SDE and thus state transition density is available, but the likelihood still remains intractable for the reason mentioned above. ...
... Since the minimisation problem above cannot be solved exactly, one can pursue a surrogate for ∆ * , in its vicinity, by minimising . Hence ∆ * , not being the true minimiser of (19), is a more conservative solution. Substituting this ∆ * into (19) gives an indication of the best relative MSE value for each C, which is of order of O (C −1 log(C)). ...
... Hence ∆ * , not being the true minimiser of (19), is a more conservative solution. Substituting this ∆ * into (19) gives an indication of the best relative MSE value for each C, which is of order of O (C −1 log(C)). In practice, we do not recommend this optimisation but rather choose (∆, η) as detailed in Section 4.1.1 and then stick to this choice even if more CPU time C has become available. ...
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Filtering Theory.- The Stochastic Process ?.- The Filtering Equations.- Uniqueness of the Solution to the Zakai and the Kushner-Stratonovich Equations.- The Robust Representation Formula.- Finite-Dimensional Filters.- The Density of the Conditional Distribution of the Signal.- Numerical Algorithms.- Numerical Methods for Solving the Filtering Problem.- A Continuous Time Particle Filter.- Particle Filters in Discrete Time.