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

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We consider inference in the scalar diffusion model dXₜ=b(Xₜ)dt+σ(Xₜ)dWₜ with discrete data (XⱼΔₙ)₀≤ ⱼ ≤ ₙ, n→ ∞, Δₙ→ 0 and periodic coefficients. For σ given, we prove a general theorem detailing conditions under which Bayesian posteriors will contract in L²-distance around the true drift function b₀ 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.