Ion channels are proteins that are located in the membranes of cells and are capable of conducting ions through the membrane. The ion channel is not always `open' for transport. The ion channel molecule may reside in several configurations, some of which correspond to an open channel and others to a closed channel. The transitions of the channel between the different configurational states have a random nature. Markov processes are often used to describe this randomness. In practice, there often exist a number of candidate Markov models. The objective of this paper is the selection of a Markov model from a finite collection of such models. We propose a Bayesian setting in which the model indicator itself is viewed as a random variable, and we develop a reversible jump Markov chain Monte Carlo (MCMC) algorithm in order to generate a sample from the posterior distribution of the model indicator given the data of a single-channel recording. A hidden Markov model is used to incorporate the correlated noise in recordings and the effects of filters that are present in the experimental set-up. The reversible jump MCMC sampler is applied to both simulated and recorded data sets.

A new method of construction of Markov chains with a given stationary distribution is proposed. The method is based on constructing an auxiliary chain with some other stationary distribution and picking elements of this auxiliary chain a suitable number of times. The proposed method is easy to implement and analyse; it may be more efficient than other related Markov chain Monte Carlo techniques. The main attractive feature of the associated Markov chain is that it regenerates whenever it accepts a new proposed point. This makes the algorithm easy to adapt and tune for practical problems. A theoretical study and numerical comparisons with some other available Markov chain Monte Carlo techniques are presented.

Let $Y$ be a stochastic process on $[0,1]$ satisfying $\rm dY(t)=n^{1/2}f(t)\rm dt + \rm dW(t)$, where $n\ge 1$ is a given scale parameter (`sample size'), $W$ is standard Brownian motion and $f$ is an unknown function. Utilizing suitable multiscale tests, we construct confidence bands for $f$ with guaranteed given coverage probability, assuming that $f$ is isotonic or convex. These confidence bands are computationally feasible and shown to be asymptotically sharp optimal in an appropriate sense.

We consider a continuous-time stochastic volatility model. The model contains a stationary volatility process, the density of which, at a fixed instant in time, we aim to estimate. We assume that we observe the process at discrete instants in time. The sampling times will be equidistant with vanishing distance. A Fourier-type deconvolution kernel density estimator based on the logarithm of the squared processes is proposed to estimate the volatility density. An expansion of the bias and a bound on the variance are derived.

Poisson shot noise is a natural generalization of a compound Poisson process when the summands are stochastic processes starting at the points of the underlying Poisson process. We study the limiting behaviour of Poisson shot noise when the limits are infinite-variance stable processes. In this context a sufficient condition for this convergence turns up which is closely related to multivariate regular variation -- we call it regular variation in the mean. We also show that the latter condition is necessary and sufficient for the weak convergence of the point processes constructed from the normalized noise sequence and also for the weak convergence of its extremes.

We show the convergence to a compound Poisson process of the high-level exceedances point process $N_n(B)= \sum_{j/n\in B} 1_{\{X_j>u_n\}}$, where $X_n=\varphi(\xi_n,Y_n) $, $ \varphi $ is a (regular) regression function, $u_n$ grows to infinity with $n$ in some suitable way, $\xi$ and $Y$ are mutually independent, $\xi$ is stationary and weakly dependent, and $Y$ is non-stationary, satisfying some ergodic conditions. The basic technique is the study of high-level exceedances of stationary processes over suitable collections of random sets.

In this paper we consider the problem of the existence of a solution for backward stochastic differential equations with uniformly continuous coefficients.

It has recently been established that the tube formula and the Euler characteristic method give an identical and valid asymptotic expansion of the tail probability of the maximum of a Gaussian random field when the random field has finite Karhunen--Lo\`eve expansion and the index set has positive critical radius. We show that the positiveness of the critical radius is an essential condition. When the critical radius is zero, we prove that only the main term is valid and that other higher-order terms are generally not valid in the formal asymptotic expansion based on the tube formula. This is done by first establishing an exact tube formula and comparing the formal tube formula with the exact formula. Furthermore, we show that the equivalence of the formal tube formula and the Euler characteristic method no longer holds when the critical radius is zero. We conclude by applying our results to some specific examples.