Abstract
We consider inference in the scalar diffusion model $\,\mathrm{d}X_{t}=b(X_{t})\,\mathrm{d}t+\sigma(X_{t})\,\mathrm{d}W_{t}$ with discrete data $(X_{j\Delta_{n}})_{0\leq j\leq n}$, $n\to\infty$, $\Delta_{n}\to0$ and periodic coefficients. For $\sigma$ given, we prove a general theorem detailing conditions under which Bayesian posteriors will contract in $L^{2}$-distance around the true drift function $b_{0}$ at the frequentist minimax rate (up to logarithmic factors) over Besov smoothness classes. We exhibit natural nonparametric priors which satisfy our conditions. Our results show that the Bayesian method adapts both to an unknown sampling regime and to unknown smoothness.
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have