Nonparametric Bayesian posterior contraction rates for discretely observed scalar diffusions

By Richard Nickl, Jakob Sohl. 2015

76 citations recorded by the source corpus.

We consider nonparametric Bayesian inference in a reflected diffusion model $dX_t = b (X_t)dt + \sigma(X_t) dW_t,$ with discretely sampled observations $X_0, X_\Delta, \dots, X_{n\Delta}$. We analyse the nonlinear inverse problem corresponding to the `low frequency sampling' regime where $\Delta>0$ is fixed and $n \to \infty$. A general theorem is proved that gives conditions for prior distributions $\Pi$ on the diffusion coefficient $\sigma$ and the drift function $b$ that ensure minimax optimal contraction rates of the posterior distribution over H\"older-Sobolev smoothness classes. These conditions are verified for natural examples of nonparametric random wavelet series priors. For the proofs we derive new concentration inequalities for empirical processes arising from discretely observed diffusions that are of independent interest.

Subfields: Nuclear and High Energy Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/2cfbbe4a3175c4a9445ca4a068f1eeaa61a2da20

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Citations in the corpus

Nonparametric statistical inference for drift vector fields of multi-dimensional diffusions (2018, result): “In this regard our results are related to recent investigations of Bayesian inverse problems [29, 39, 4, 17, 28, 35], Bernstein-von Mises theorems [40, 32, 31, 33] and diffusion models [50, 38, 54, 34, 49, 26, 1].”