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

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

Use of Input Deformations with Brownian Motion Filters for Discontinuous Regression

Abstract. Bayesian Gaussian processes are known as ‘smoothing devices’ and in the case of n data points they require O(n 2)... O(n 3) number of multiplications in order to perform a regression analysis. In this work we consider one-dimensional regression with Wiener-Lévy (Brownian motion) covariance functions. We indicate that they require only O(n) number of multiplications and show how one can utilize input deformations in order to define a much broader class of efficient covariance functions suitable for discontinuity-preserving filtering. An example of the selective smoothing is presented which shows that regression with Brownian motion filters outperforms or improves nonlinear diffusion filtering especially when observations are contaminated with noise of larger variance.