Nonparametric Bayesian estimation of a H\"older continuous diffusion coefficient

TL;DR

This paper introduces a Bayesian nonparametric method using histogram priors with inverse Gamma distributions to estimate diffusion coefficients in stochastic differential equations, achieving optimal convergence rates and practical effectiveness.

Contribution

It proposes a novel histogram-based Bayesian approach with inverse Gamma priors for diffusion coefficient estimation, providing theoretical guarantees and practical implementation.

Findings

Posterior contraction rate is optimal for H"older continuous coefficients.

Method is straightforward to implement with inverse Gamma posteriors.

Practical results demonstrate effectiveness on real exchange rate data.

Abstract

We consider a nonparametric Bayesian approach to estimate the diffusion coefficient of a stochastic differential equation given discrete time observations over a fixed time interval. As a prior on the diffusion coefficient, we employ a histogram-type prior with piecewise constant realisations on bins forming a partition of the time interval. Specifically, these constants are realizations of independent inverse Gamma distributed randoma variables. We justify our approach by deriving the rate at which the corresponding posterior distribution asymptotically concentrates around the data-generating diffusion coefficient. This posterior contraction rate turns out to be optimal for estimation of a H\"older-continuous diffusion coefficient with smoothness parameter $0<\lambda\leq 1.$ Our approach is straightforward to implement, as the posterior distributions turn out to be inverse Gamma again, and leads to good practical results in a wide range of simulation examples. Finally, we apply our method on exchange rate data sets.