Nonparametric estimation of diffusions: a differential equations approach

Abstract

We consider estimation of scalar functions which determine the dynamics of diffusion processes. It has been recently shown that nonparametric maximum likelihood is ill-posed in this context. We adopt a probabilistic approach to regularize the problem by the adoption of a prior distribution for the unknown functional. A Gaussian prior measure is specified in the function space by means of its precision operator, which is defined as an appropriate differential operator. We establish that a Bayesian Gaussian conjugate analysis for the drift of one-dimensional non-linear diffusions is feasible given high-frequency data. This is achieved by expressing the log-likelihood as a quadratic function of the drift, with sufficient statistics given by the so-called local time process and the end points of the observed path. Computationally efficient posterior inference is carried out using a finite element method. We embed this technology in partially observed situations and adopt a data augmentation approach whereby we iteratively generate missing data paths and draws from the unknown functional. Our methodology is applied to estimate the drift of models used in molecular dynamics and financial econometrics using high and low frequency observations. We discuss extensions to other partially observed schemes and connections to other types of non-parametric inference.

Sparsity-promoting and edge-preserving maximum a posteriori estimators in non-parametric Bayesian inverse problems

“…First, by (41), G(u n ) → G(u † ) in probability as n → ∞. Therefore there exists a subsequence which satisfies (after labelling by n again) G(u n ) → G(u † ) almost surely.…”

The nonparametric LAN expansion for discretely observed diffusions

“…and (25) holds. The next proposition regarding the existence, uniqueness and regularity properties of solutions to (25) will play a key role for the proofs in the rest of Section 3. To state the result, we need the following interpolation spaces D(α), 0 ≤ α ≤ 1, between L 2 and D:…”

Variational Inference for Stochastic Differential Equations

“…This is relatively easy, when the data is free of noise and is observed at high frequency. [6] This means, we can assume that the (for simplicity constant) time τ between observations is small enough such that the discretized dynamics (2) applies to consecutive data points. Hence, the data is assumed to be generated as…”

“…[7] While a large amount of data makes the necessary inversions of large matrices nontrivial, a tractable solution is possible for the case, where K −1 is a differential operator. [6] The practical inversion of such large linear systems can be achieved with implicit schemes, see ref. [8].…”

“…This is possible, for example, within a Monte Carlo sampling approach. Papaspiliopoulos et al [6] have developed a Gibbs sampler which iteratively updates samples from SDE paths and from drift functions. Using Bayes rule again, one can represent the conditional path measure as P(X 0:T |y, f ) = P(X 0:T | f )P(y|X 0:T ) P(y| f )…”