Nonparametric Bayesian posterior contraction rates for scalar diffusions with high-frequency data

Understand

We consider inference in the scalar diffusion model $dX_t=b(X_t)dt+\sigma(X_t)dW_t$ with discrete data $(X_{j\Delta_n})_{0\leq j \leq n}$, $n\to \infty,~\Delta_n\to 0$ 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.