2018/07/11 by Palle E. T. Jørgensen, Feng Tian, Jorgensen, Palle +1
Economics, Econometrics and Finance · #22E70 #31A15 #31C20 #39A12 #42C15 #46N30 #46N50 #58J65 #62D05 #65R10 #94A20 #Complex Systems and Time Series Analysis #FOS: Mathematics #Financial Risk and Volatility Modeling #Functional Analysis (math.FA) #Primary 47L60 #Probability (math.PR) #Secondary 46N20 #Stochastic processes and financial applications
paper · pdf · doi:10.48550/arxiv.1807.04111
openalex publication_date 2018/07/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Starting with the correspondence between positive definite kernels on the one hand and reproducing kernel Hilbert spaces (RKHSs) on the other, we turn to a detailed analysis of associated measures and Gaussian processes. Point of departure: Every positive definite kernel is also the covariance kernel of a Gaussian process. Given a fixed sigma-finite measure μ, we consider positive definite kernels defined on the subset of the sigma algebra having finite μ measure. We show that then the corresponding Hilbert factorizations consist of signed measures, finitely additive, but not automatically sigma-additive. We give a necessary and sufficient condition for when the measures in the RKHS, and the Hilbert factorizations, are sigma-additive. Our emphasis is the case when μ is assumed non-atomic. By contrast, when μ is known to be atomic, our setting is shown to generalize that of Shannon-interpolation. Our RKHS-approach further leads to new insight into the associated Gaussian processes, their Itô calculus and diffusion. Examples include fractional Brownian motion, and time-change processes.