COMPUTATIONAL DOOB'S h-TRANSFORMS FOR ON-LINE FILTERING OF DISCRETELY OBSERVED DIFFU-SIONS

Abstract

This paper is concerned with online filtering of discretely observed nonlinear diffusion processes. Our approach is based on the fully adapted auxiliary particle filter, which involves Doob's h-transforms that are typically intractable. We propose a computational framework to approximate these h-transforms by solving the underlying backward Kolmogorov equations using nonlinear Feynman-Kac formulas and neural networks. The methodology allows one to train a locally optimal particle filter prior to the data-assimilation procedure. Numerical experiments illustrate that the proposed approach can be orders of magnitude more efficient than state-of-the-art particle filters in the regime of highly informative observations, when the observations are extreme under the model, and if the state dimension is large.

1. INTRODUCTION

Diffusion processes are fundamental tools in applied mathematics, statistics, and machine learning. Because this flexible class of models is easily amenable to computations and simulations, diffusion processes are very common in biological sciences (e.g. population and multi-species models, stochastic delay population systems), neuroscience (e.g. models for synaptic input, stochastic Hodgkin-Huxley model, stochastic Fitzhugh-Nagumo model), and finance (e.g. modeling multi assets prices) (Allen, 2010; Shreve et al., 2004; Capasso & Capasso, 2021) . In these disciplines, tracking signal from partial or noisy observations is a very common task. However, working with diffusion processes can be challenging as their transition densities are only tractable in rare and simple situations such as (geometric) Brownian motions or Ornstein-Uhlenbeck (OU) processes. This difficulty has hindered the use of standard methodologies for inference and data-assimilation of models driven by diffusion processes and various approaches have been developed to circumvent or mitigate some of these issues, as discussed in Section 4. Consider a time-homogeneous multivariate diffusion process dX t = µ(X t ) dt + σ(X t ) dB t that is discretely observed at regular intervals. Noisy observations y k of the latent process X t k are collected at equispaced times t k ≡ k T for k ≥ 1. We consider the online filtering problem which consists in estimating the conditional laws π k (dx) = P(X t k ∈ dx|y 1 , . . . , y k ), i.e. the filtering distributions, as observations are collected. We focus on the use of Particle Filters (PFs) that approximate the filtering distributions with a system of weighted particles. Although many previous works have relied on the Bootstrap Particle Filter (BPF), which simulates particles from the diffusion process, it can perform poorly in challenging scenarios as it fails to take the incoming observation y k into account. The goal of this article is to show that the (locally) optimal approach given by the Fully Adapted Auxiliary Particle Filter (FA-APF) (Pitt & Shephard, 1999) can be implemented. This necessitates simulating a conditioned diffusion process, which can be formulated as a control problem involving an intractable Doob's h-transform (Rogers & Williams, 2000; Chung & Walsh, 2006) . We propose the Computational Doob's h-Transform (CDT) framework for efficiently approximating these quantities. The method relies on nonlinear Feynman-Kac formulas for solving backward Kolmogorov equations simultaneously for all possible observations. Importantly, this preprocessing step only needs to be performed once before starting the online filtering procedure. Numerical experiments illustrate that the proposed approach can be orders of magnitude more efficient than the BPF in the regime of highly informative observations, when the observations are extreme under the model, and

