Inference for diffusions from low frequency measurements
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Let (X_t) be a reflected diffusion process in a bounded convex domain in ℝ^d, solving the stochastic differential equation dX_t = ∇ f(X_t) dt + √(2f (X_t)) dW_t, t ≥ 0, with W_t a d-dimensional Brownian motion. The data X_0, X_D, …, X_ND consist of discrete measurements and the time interval D between consecutive observations is fixed so that one cannot `zoom' into the observed path of the process. The goal is to solve the non-linear inverse problem of inferring the diffusivity f and the associated transition operator P_t,f. We prove injectivity theorems and stability estimates for the maps f ↦ P_t,f↦ P_D,f, t<D. Using these estimates we then establish the statistical consistency of a class of Bayesian algorithms based on Gaussian process priors for the infinite-dimensional parameter f, and show optimality of some of the convergence rates obtained. We discuss an underlying relationship between `fast convergence' and the `hot spots' conjecture from spectral geometry.
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