2014/10/02 by Maik Görgens, Görgens, Maik, Ingemar Kaj +1 · 1 citation
Economics, Econometrics and Finance · Mathematics · #Complex Systems and Time Series Analysis #Financial Risk and Volatility Modeling #Stochastic processes and financial applications #math.PR #msc:60G15 #msc:60G60
paper · pdf · doi:10.48550/arxiv.1410.0511
26 pages
arxiv created 2014/10/02 · arxiv updated 2014/10/03
We consider the class of selfsimilar Gaussian generalized random fields introduced by Dobrushin in 1979. These fields are indexed by Schwartz functions on ℝd and parametrized by a self-similarity index and the degree of stationarity of their increments. We show that such Gaussian fields arise in explicit form by letting Gaussian white noise, or Gaussian random balls white noise, drive a shift and scale shot-noise mechanism on ℝd, covering both isotropic and anisotropic situations. In some cases these fields allow indexing with a wider class of signed measures, and by using families of signed measures parametrized by the points in euclidean space we are able to extract pointwise defined Gaussian processes, such as fractional Brownian motion on ℝd. Developing this method further, we construct Gaussian bridges and Gaussian membranes on a finite domain, which vanish on the boundary of the domain.