2015/12/03 by Milan Stehlik, Stehlik, Milan, Christian Helpersdorfer +3
Economics, Econometrics and Finance · Engineering · Mathematics · #FOS: Computer and information sciences #Methodology (stat.ME) #Reservoir Engineering and Simulation Methods #Statistical and numerical algorithms #Stochastic processes and financial applications #stat.ME
paper · pdf · doi:10.48550/arxiv.1512.01257
arxiv created 2015/12/03 · openalex publication_date 2015/12/03 · arxiv updated 2015/12/07 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28
A.N. Kolmogorov proposed several problems on stochastic processes, which has been rarely addressed later on. One of the open problems are stochastic processes with discontinuous covariance function. For example, semicontinuous covariance functions have been used in regression and kriging by many authors in statistics recently. In this paper we introduce purely topologically defined regularity conditions on covariance kernels which are still applicable for increasing and infill domain asymptotics for regression problems, kriging and finance. These conditions are related to semicontinuous maps of Ornstein Uhlenbeck (OU) processes. Beside this new regularity conditions relax the continuity of covariance function by consideration of semicontinuous covariance. We provide several novel applications of the introduced class for optimal design of random fields, random walks in finance and probabilities of ruins related to shocks, e.g. by earthquakes. In particular we construct a random walk model with semicontinuous covariance.