Posterior contraction rate for non-parametric Bayesian estimation of the dispersion coefficient of a stochastic differential equation

Abstract

Abstract. We derive the posteror contraction rate for non-parametric Bayesian estimation of a deterministic dispersion coefficient of a linear stochastic differential equation.

Nonparametric Bayesian Volatility Estimation

“…Literature on nonparametric Bayesian volatility estimation in SDE models is scarce. We can list theoretical contributions (Gugushvili and Spreij, 2014a), (Gugushvili and Spreij, 2016), (Nickl and Söhl, 2017), and the practically oriented paper (Batz et al, 2017feb). The model in the former two papers is close to the one considered in the present work, but from the methodological point of view different Bayesian priors are used and practical usefulness of the corresponding Bayesian approaches is limited.…”

Nonparametric Bayesian Estimation of a HHlder Continuous Diffusion Coefficient

“…Denote S n (s) = n −1 log R n (s) and note that D n = Sn exp(nS n (s))Π n (ds). As in Gugushvili and Spreij (2016), we write…”

“…For parametric approaches to inference in SDE models, see, e.g., Chapter 2 in Kutoyants (2004), Chapter 3 in Iacus (2008), and references therein. Nonparametric statistical inference for SDEs of the type studied in the present work has been considered in Genon-Catalot et al (1992), Hoffmann (1997) and Soulier (1998) within the frequentist setup, while and Gugushvili and Spreij (2016) have explored the problem from the Bayesian perspective. Although the nonparametric methods these papers study are implementable in principle, these works are primarily of theoretical nature and practical performance of the corresponding approaches is not clear.…”

“…Although the nonparametric methods these papers study are implementable in principle, these works are primarily of theoretical nature and practical performance of the corresponding approaches is not clear. Furthermore, except and Gugushvili and Spreij (2016), there is hardly any other work available on estimation of the dispersion coefficient (or diffusion coefficient) from the nonparametric Bayesian point of view, which constitutes the central topic of our paper. In this context we can mention only a theoretical contribution Nickl and Söhl (2017) and a practically oriented paper Batz et al (2018), but the models considered there, as well as the sampling scheme, are different from ours, and the theory developed in Nickl and Söhl (2017) does not cover the approach in Batz et al (2018).…”

“…The main practical challenges for Bayesian inference in SDE models from discrete observations are an intractable likelihood and absence of a closed form expression for the posterior distribution, which complicates considerably the inference; see, e.g., Roberts and Stramer (2001), Elerian et al (2001), Fuchs (2013) and van der Meulen and Schauer (2017). We circumvent these difficulties by intentionally misspecifying the drift coefficient, and employing a (conjugate) histogramtype prior on the diffusion coefficient, that has piecewise constant realisations on bins forming a partition of [0, T ] (this is different from and Gugushvili and Spreij (2016), where the drift b 0 is in fact zero, and other priors are used). Due to this, our nonparametric Bayesian method to estimate the dispersion coefficient s 0 in (1) is easily implemented, fast and requires little fine-tuning from the user.…”