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By using a simple observation that the density processes appearing in Ito's martingale representation theorem are invariant under the change of measures, we establish a non-linear version of the Cameron-Martin formula for solutions of a…
We present a notion of almost periodicity wich can be applied to random dynamical systems as well as almost periodic stochastic differential equations in Hilbert spaces (abstract stochastic partial differential equations). This concept…
We introduce a new tool for the quantitative characterisation of the departure form Markovianity of a given dynamical process. Our tool can be applied to a generic $N$-level system and extended straightforwardly to Gaussian…
In this paper we introduce a novel online time series forecasting model we refer to as the pM-GP filter. We show that our model is equivalent to Gaussian process regression, with the advantage that both online forecasting and online…
This paper is concerned with statistical inference for infinite range interaction Gibbs point processes and in particular for the large class of Ruelle superstable and lower regular pairwise interaction models. We extend classical…
This paper establishes the strong consistency and asymptotic normality of the quasi-maximum likelihood estimator (QMLE) for a GARCH process with periodically time-varying parameters. We first give a necessary and sufficient condition for…
Recently, a class of stochastic processes known as piecewise deterministic Markov processes has been used to define continuous-time Markov chain Monte Carlo algorithms with a number of attractive properties, including compatibility with…
Self-similarity of systems is very popular and intensively developing field during last decades. To this field belong so-called stable distributions and their generalization. In Klebanov and Sl\'amov\'a (2014) there was given an approach to…
Fitting regression models for intensity functions of spatial point processes is of great interest in ecological and epidemiological studies of association between spatially referenced events and geographical or environmental covariates.…
Hybrid systems whose mode dynamics are governed by non-linear ordinary differential equations (ODEs) are often a natural model for biological processes. However such models are difficult to analyze. To address this, we develop a…
Continuous-time Markov processes over finite state-spaces are widely used to model dynamical processes in many fields of natural and social science. Here, we introduce an maximum likelihood estimator for constructing such models from data…
We examine an analytic variational inference scheme for the Gaussian Process State Space Model (GPSSM) - a probabilistic model for system identification and time-series modelling. Our approach performs variational inference over both the…
Piecewise-deterministic Markov processes (PDMPs) are often used to model abrupt changes in the global environment or capabilities of a controlled system. This is typically done by considering a set of "operating modes" (each with its own…
In many spatial and spatial-temporal models, and more generally in models with complex dependencies, it may be too difficult to carry out full maximum likelihood (ML) analysis. Remedies include the use of pseudo-likelihood (PL) and…
In this paper we investigate quasi-stationary distributions {\mu}_N of stochastic approximation algorithms with constant step size which can be viewed as random perturbations of a time-continuous dynamical system. Inspired by ecological…
We develop Markov categories as a framework for synthetic probability and statistics, following work of Golubtsov as well as Cho and Jacobs. This means that we treat the following concepts in purely abstract categorical terms: conditioning…
Standard GPs offer a flexible modelling tool for well-behaved processes. However, deviations from Gaussianity are expected to appear in real world datasets, with structural outliers and shocks routinely observed. In these cases GPs can fail…
This survey describes the method of approximation of operator semigroups, based on the Chernoff theorem. We outline recent results in this domain as well as clarify relations between constructed approximations, stochastic processes,…
Extreme-value theory for random vectors and stochastic processes with continuous trajectories is usually formulated for random objects all of whose univariate marginal distributions are identical. In the spirit of Sklar's theorem from…
Continuous input signals like images and time series that are irregularly sampled or have missing values are challenging for existing deep learning methods. Coherently defined feature representations must depend on the values in unobserved…