Moritz Schauer

Reversible jump MCMC for nonparametric drift estimation for diffusion processes

In the context of nonparametric Bayesian estimation a Markov chain Monte Carlo algorithm is devised and implemented to sample from the posterior distribution of the drift function of a continuously or discretely observed one-dimensional diffusion. The drift is modeled by a scaled linear combination of basis functions with a Gaussian prior on the coefficients. The scaling parameter is equipped with a partially conjugate prior. The number of basis functions in the drift is equipped with a prior distribution as well. For continuous data, a reversible jump Markov chain algorithm enables the exploration of the posterior over models of varying dimension. Subsequently, it is explained how data-augmentation can be used to extend the algorithm to deal with diffusions observed discretely in time. Some examples illustrate that the method can give satisfactory results. In these examples a comparison is made with another existing method as well.

Guided proposals for simulating multi-dimensional diffusion bridges

A Monte Carlo method for simulating a multi-dimensional diffusion process conditioned on hitting a fixed point at a fixed future time is developed. Proposals for such diffusion bridges are obtained by superimposing an additional guiding term to the drift of the process under consideration. The guiding term is derived via approximation of the target process by a simpler diffusion processes with known transition densities. Acceptance of a proposal can be determined by computing the likelihood ratio between the proposal and the target bridge, which is derived in closed form.We show under general conditions that the likelihood ratio is well defined and show that a class of proposals with guiding term obtained from linear approximations fall under these conditions.

Network Coloring and Colored Coin Games

Kearns et al. introduced the Graph Coloring Problem to model dynamic conflict resolution in social networks. Players, represented by the nodes of a graph, consecutively update their color from a fixed set of colors with the prospect of finally choosing a color that differs from all neighbors choices. The players only react on local information (the colors of their neighbors) and do not communicate. The reader might think of radio stations searching for transmission frequencies which are not subject to interference from other stations. While Kearns et al. (see [10]) empirically examined how human players deal with such a situation, Chaudury et al. performed a theoretical study and showed that, under a simple, greedy and selfish strategy, the players find a proper coloring of the graph within time \(O\left (\log \left (\frac{n} {\delta } \right )\right )\) with probability ≥ 1 − δ, where n is the number of nodes in the network and δ is arbitrarily small. In other words, the graph is properly colored within τ steps and \(\tau < c\log \left (\frac{n} {\delta } \right )\) with high probability for some constant c. Previous estimates on the constant c are very large. In this chapter we substantially improve the analysis and upper time bound for the proper coloring, by combining ideas from search games and probability theory.

Bayesian estimation of discretely observed multi-dimensional diffusion processes using guided proposals

Estimation of parameters of a diffusion based on discrete time observations poses a difficult problem due to the lack of a closed form expression for the likelihood. From a Bayesian computational perspective it can be casted as a missing data problem where the diffusion bridges in between discrete-time observations are missing. The computational problem can then be dealt with using a Markov-chain Monte-Carlo method known as data-augmentation. If unknown parameters appear in the diffusion coefficient, direct implementation of data-augmentation results in a Markov chain that is reducible. Furthermore, data-augmentation requires efficient sampling of diffusion bridges, which can be difficult, especially in the multidimensional case. We present a general framework to deal with with these problems that does not rely on discretisation. The construction generalises previous approaches and sheds light on the assumptions necessary to make these approaches work. We define a random-walk type Metropolis-Hastings sampler for updating diffusion bridges. Our methods are illustrated using guided proposals for sampling diffusion bridges. These are Markov processes obtained by adding a guiding term to the drift of the diffusion. We give general guidelines on the construction of these proposals and introduce a time change and scaling of the guided proposal that reduces discretisation error. Numerical examples demonstrate the performance of our methods.

Bayesian estimation of incompletely observed diffusions

Abstract We present a general framework for Bayesian estimation of incompletely observed multivariate diffusion processes. Observations are assumed to be discrete in time, noisy and incomplete. We assume the drift and diffusion coefficient depend on an unknown parameter. A data-augmentation algorithm for drawing from the posterior distribution is presented which is based on simulating diffusion bridges conditional on a noisy incomplete observation at an intermediate time. The dynamics of such filtered bridges are derived and it is shown how these can be simulated using a generalised version of the guided proposals introduced in Schauer, Van der Meulen and Van Zanten (2017, Bernoulli 23(4A)).

Adaptive nonparametric drift estimation for diffusion processes using Faber--Schauder expansions

We consider the problem of nonparametric estimation of the drift of a continuously observed one-dimensional diffusion with periodic drift. Motivated by computational considerations, van der Meulen et al. (Comput Stat Data Anal 71:615–632, 2014) defined a prior on the drift as a randomly truncated and randomly scaled Faber–Schauder series expansion with Gaussian coefficients. We study the behaviour of the posterior obtained from this prior from a frequentist asymptotic point of view. If the true data generating drift is smooth, it is proved that the posterior is adaptive with posterior contraction rates for the

Simulation of elliptic and hypo-elliptic conditional diffusions

L_2

Bayesian wavelet de-noising with the caravan prior.

L2-norm that are optimal up to a log factor. Contraction rates in