Optimal design, financial and risk modelling with stochastic processes having semicontinuous covariances

TL;DR

This paper introduces topologically defined regularity conditions for semicontinuous covariance functions in stochastic processes, enabling new applications in optimal design, finance, and risk modeling.

Contribution

It proposes novel regularity conditions for semicontinuous covariances, extending stochastic process modeling beyond traditional continuous covariances.

Findings

Developed topological regularity conditions for semicontinuous covariances

Applied these conditions to optimal design and financial risk models

Constructed a new random walk model with semicontinuous covariance

Abstract

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.