Self-dual continuous processes

The important application of semi-static hedging in financial markets naturally leads to the notion of conditionally quasi self-dual processes which is, for continuous semimartingales, related to conditional symmetry properties of both their ordinary as well as their stochastic logarithms. We provide a structure result for continuous conditionally quasi self-dual processes. Our main result is to give a characterization of continuous Ocone martingales via a strong version of self-duality.

Stationarity of multivariate particle systems

A particle system is a family of i.i.d. stochastic processes with values translated by Poisson points. We obtain conditions that ensure the stationarity in time of the particle system in RdRd and in some cases provide a full characterisation of the stationarity property. In particular, a full characterisation of stationary multivariate Brown–Resnick processes is given.

Temporal Changes in Randomness of Bird Communities across Central Europe

Many studies have examined whether communities are structured by random or deterministic processes, and both are likely to play a role, but relatively few studies have attempted to quantify the degree of randomness in species composition. We quantified, for the first time, the degree of randomness in forest bird communities based on an analysis of spatial autocorrelation in three regions of Germany. The compositional dissimilarity between pairs of forest patches was regressed against the distance between them. We then calculated the y-intercept of the curve, i.e. the ‘nugget’, which represents the compositional dissimilarity at zero spatial distance. We therefore assume, following similar work on plant communities, that this represents the degree of randomness in species composition. We then analysed how the degree of randomness in community composition varied over time and with forest management intensity, which we expected to reduce the importance of random processes by increasing the strength of environmental drivers. We found that a high portion of the bird community composition could be explained by chance (overall mean of 0.63), implying that most of the variation in local bird community composition is driven by stochastic processes. Forest management intensity did not consistently affect the mean degree of randomness in community composition, perhaps because the bird communities were relatively insensitive to management intensity. We found a high temporal variation in the degree of randomness, which may indicate temporal variation in assembly processes and in the importance of key environmental drivers. We conclude that the degree of randomness in community composition should be considered in bird community studies, and the high values we find may indicate that bird community composition is relatively hard to predict at the regional scale.

Stochastic search for semiparametric linear regression models

This paper introduces and analyzes a stochastic search method for parameter estimation in linear regression models in the spirit of Beran and Millar [Ann. Statist. 15(3) (1987) 1131–1154]. The idea is to generate a random finite subset of a parameter space which will automatically contain points which are very close to an unknown true parameter. The motivation for this procedure comes from recent work of Dümbgen et al. [Ann. Statist. 39(2) (2011) 702–730] on regression models with log-concave error distributions.

Quasi-Self-Dual Exponential Lévy Processes

The important application of semistatic hedging in financial markets naturally leads to the notion of quasi--self-dual processes. The focus of our study is to give new characterizations of quasi--self-duality. We analyze quasi--self-dual Lévy driven markets which do not admit arbitrage opportunities and derive a set of equivalent conditions for the stochastic logarithm of quasi--self-dual martingale models. Since for nonvanishing order parameter two martingale properties have to be satisfied simultaneously, there is a nontrivial relation between the order and shift parameter representing carrying costs in financial applications. This leads to an equation containing an integral term which has to be inverted in applications. We first discuss several important properties of this equation and, for some well-known Lévy-driven models, we derive a family of closed-form inversion formulae.