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 if the state dimension is large. A PyTorch implementation to reproduce our numerical experiments is available at https://anonymous.4open.science/r/CompDoobTransform/.

Notations. For two matrices

A, B ∈ R d,d , their Frobenius inner product is defined as ⟨A, B⟩ F = d i,j=1 A i,j B i,j . The Euclidean inner product for u, v ∈ R d is denoted as ⟨u, v⟩ = d i=1 u i v i . For two (or more) functions F and G, we sometimes use the shortened notation [F G](x) to denote the product F (x)G(x).

2. BACKGROUND 2.1 FILTERING OF DISCRETELY OBSERVED DIFFUSIONS

Consider a homogeneous diffusion process {X t } t≥0 in X = R d with initial distribution ρ 0 (dx) and dynamics  dX t = µ(X t ) dt + σ(X t ) dB t , (Y k ∈ A | X t k = x k ) = A g(x k , y) dy for some dominating measure dy on Y. For a test function φ : X → R, the generator of the diffusion process {X t } t≥0 is given by Lφ = ⟨µ, ∇φ⟩ + 1 2 ⟨σσ ⊤ , ∇ 2 φ⟩ F . This article is concerned with approximating the filtering distributions π k (dx) = P(X t k ∈ dx | y 1 , . . . , y k ). For notational convenience, we set π 0 (dx) ≡ ρ 0 (dx) since there is no observation collected at the initial time t = 0.

2.2. PARTICLE FILTERING

Particle Filters (PF), also known as Sequential Monte Carlo methods, are a set of Monte Carlo algorithms that can be used to solve filtering problems (see Chopin et al. (2020) for a recent textbook on the topic). PFs evolve a set of M ≥ 1 particles x 1:M t = (x 1 t , . . . , x M t ) ∈ X M forward in time using a combination of propagation and resampling operations. To initialize the PF, each initial particle x j 0 ∈ X for 1 ≤ j ≤ M is sampled independently from the distribution ρ 0 (dx) so that π 0 (dx) ≈ M -1 M j=1 δ(dx; x j 0 ). Approximations of the filtering distribution π k for k ≥ 1 are built recursively as follows. Given the Monte Carlo approximation of the filtering distribution at time t k , π k (dx) ≈ M -1 M j=1 δ(dx; x j t k ), the particles x 1:M t k are propagated independently forward in time by x j t k+1 ∼ q k+1 (d x | x j t k ), using a Markov kernel q k+1 (d x | x) specified by the user. The BPF corresponds to the Markov kernel q BPF k+1 (d x | x) = P(X t k+1 ∈ d x | X t k = x), while the FA-APF (Pitt & Shephard, 1999) corresponds to the (typically intractable) kernel q FA-APF k+1 (d x | x) = P(X t k+1 ∈ d x | X t k = x, Y k+1 = y k+1 ). Each particle x j t k+1 is associated with a normalized weight W j k+1 = W j k+1 / M i=1 W i k+1 , where the unnormalized weights W j k+1 (by time-homogeneity of (1)) are defined as W j k+1 = p T (d x j t k+1 | x j t k ) q k+1 (d x j t k+1 | x j t k ) g( x j t k+1 , y k+1 ). (2) The BPF and FA-APF correspond respectively to having W j,BPF k+1 = g( x j t k+1 , y k+1 ) and W j,FA-APF k+1 = E[g(X t k+1 , y k+1 ) | X t k = x j t k ]. (3) The weights are such that π k+1 (dx) ≈ M j=1 W j k+1 δ(dx; x j t k+1 ). The resampling step consists in defining a new set of particles x 1:M t k+1 with P(x j t k+1 = x i t k+1 ) = W i k+1 . This resampling scheme ensures that the equally weighted set of particles x 1:M t k+1 provides a Monte Carlo approximation of the

