MEXIT: Maximal un-coupling times for stochastic processes

Classical coupling constructions arrange for copies of the same Markov process started at two different initial states to become equal as soon as possible. In this paper, we consider an alternative coupling framework in which one seeks to arrange for two different Markov (or other stochastic) processes to remain equal for as long as possible, when started in the same state. We refer to this “un-coupling” or “maximal agreement” construction as MEXIT, standing for “maximal exit”. After highlighting the importance of un-coupling arguments in a few key statistical and probabilistic settings, we develop an explicit MEXIT construction for stochastic processes in discrete time with countable state-space. This construction is generalized to random processes on general state-space running in continuous time, and then exemplified by discussion of MEXIT for Brownian motions with two different constant drifts.     

A measure of mutual complete dependence in discrete variables through subcopula

Siburg and Stoimenov [12] gave a measure of mutual complete dependence of continuous variables which is different from Spearman's ρ and Kendall's τ. In this paper, a similar measure of mutual complete dependence is applied to discrete variables. Also two measures for functional relationships, which are not bijection, are investigated. For illustration of our main results, several examples are given.     

Spaces of idempotent measures of compact metric spaces

We investigate certain geometric properties of the spaces of idempotent measures. In particular, we prove that the space of idempotent measures on an infinite compact metric space is homeomorphic to the Hilbert cube.     

Integral Self-affiine Tiles of Bandt＇s Model

Integral self-affine tiling of Bandt＇s model is a generalization of the integral self-affine tiling. Using ergodic theory, we show that the Lebesgue measure of the tile is a rational number where the denominator equals to the order of the associate symmetry group. We apply the result to the study of the Levy Dragon.     

Optimal risk transfer for agents with germs

We introduce a new class of risk measures called generalized entropic risk measures (GERMS) that allow economic agents to have different attitudes towards different sources of risk. We formulate the problem of optimal risk transfer in terms of these risk measures and characterize the optimal transfer contract. The optimal contract involves what we call intertemporal source-dependent quotient sharing, where agents linearly share changes in the aggregate risk reserve that occur in response to shocks to the system over time, with scaling coefficients that depend on the attitudes of each agent towards the source of risk causing the shock. Generalized entropic risk measures are not dilations of a common base risk measure, so our results extend the class of risk measures for which explicit characterizations of the optimal transfer contract can be found.     

Level Sets of Multiparameter Stable Processes

We establish the correct Hausdorff measure function for the level sets of additive strictly stable processes derived from strictly stable processes satisfying Taylor’s condition (A). This leads naturally to a characterization of local time in terms of the corresponding Hausdorff measure function of the level set.      

Time Regularity for Random Walks on Locally Compact Groups

Let G be a compactly generated, locally compact group, and let T be the operator of convolution with a probability measure μ on G. Our main results give sufficient conditions on μ for the operator T to be analytic in L ^p (G), 1 < p < ∞, where analyticity means that one has an estimate of form for all n = 1, 2, ... in L ^p operator norm. Counterexamples show that analyticity may not hold if some of the conditions are not satisfied.