A Multiresolution Method for Parameter Estimation of Diffusion Processes

Abstract

Diffusion process models are widely used in science, engineering and finance. Most diffusion processes are described by stochastic differential equations in continuous time. In practice, however, data is typically only observed at discrete time points. Except for a few very special cases, no analytic form exists for the likelihood of such discretely observed data. For this reason, parametric inference is often achieved by using discrete-time approximations, with accuracy controlled through the introduction of missing data. We present a new multiresolution Bayesian framework to address the inference difficulty. The methodology relies on the use of multiple approximations and extrapolation, and is significantly faster and more accurate than known strategies based on Gibbs sampling. We apply the multiresolution approach to three data-driven inference problems – one in biophysics and two in finance – one of which features a multivariate diffusion model with an entirely unobserved component.

Bayesian analysis of ambulatory blood pressure dynamics with application to irregularly spaced sparse data

“…The acceptance rates of the cross-resolution move are 0.04, 0.14 and 0.27 at the 2nd, 3rd and 4th resolution, respectively. The increasing acceptance rates agree with Kou et al (2012). As k increases, the empirical distribution for P k−1 (θ, X (k−1) |Y) becomes a better proposal distribution because the difference between P k−1 (θ, X (k−1) |Y) and P k (θ, X (k) |Y) decreases.…”

“…Due to the intractability of the proposed SDE, we employ discretization approximation [Pedersen (1995)]. Unfortunately, real-life ambulatory CV data are characterized by much sparser and irregularly spaced time intervals than those investigated in most simulation studies involving nonlinear SDE models [Kou et al (2012), Lindström (2012)]. Achieving reasonable estimation properties necessitates the use of a large number of imputations between subsequent observed intervals, a procedure that quickly becomes inefficient for the kind of data considered.…”

A New Framework for Inference on Markov Population Models

“…In Section 3 we showed that Markov population models can be approximated by the system of SDEs in (6). Performing inference on θ in the absence of an analytic solution consists of two steps; first ( 6) is approximated by a numerical method, second an approximate likelihood is constructed from the numerical solution (Doucet and Johansen, 2009;Kou et al, 2012). In this section, we introduce the Euler-Maruyama scheme, which is the most commonly used method for approximating (6) (Sun et al, 2015;Allen, 2017).…”

Bayesian Multiresolution Variable Selection for Ultra-High Dimensional Neuroimaging Data

“…From the computational perspective, Giles [33] showed that the computational complexity for estimating the expected value from a stochastic differential equation could be reduced by a multiresolution Monte Carlo simulation. More recently, Kou et al [34] applied a multiresolution method to diffusion process models for discrete data and showed that their approach improves computational efficiency and estimation accuracy. From the perspective of model construction, Fox and Dunson [35] adopted the multiresolution idea in Gaussian process models to capture both long-range dependencies and abrupt discontinuities.…”

“…In this work, we develop efficient multiresolution MCMC algorithms for variable selection in the ultra-high dimensional feature space of imaging data. In contrast to the coupled Markov chain methods [30], [31], [34] that alternate between different resolutions in posterior simulation, we construct and conduct posterior computations for a sequence of nested auxiliary models for variable selection from the coarsest scale to the finest scale. Our goal is to conduct variable selection at the finest scale – the resolution in the observed data.…”