On mixed exponential processes and martingales
type: Item
instance of: scholarly article
title: On mixed exponential processes and martingales
zbMATH Open document ID: 0944.60051
DOI: 10.1007/BF02465543
published in: Lithuanian Mathematical Journal
publication date: 2000-09-18T00:00:00Z
review text: The author considers Hilbert space \(H\)-valued sequences of independent random variables and their partial sums \((X_n)\). On the product space of realizations the natural exponential family of measures \((P^\theta)\) is constructed. The mixed distribution \(P^U:= \int_\Theta P^\theta(d\theta)\) is studied. \((X_n)\) is Markovian under \(P^U\). The Radon-Nikodým-derivative \({dP^U_n\over dP^0_n}= q^U_n(X_n)\) is calculated, where \(q^U_n(y)\) is a function on \(N\times H\) being regular for the space time process \((n,X_n)\). All mixed distributions \(P^U\) are characterized by this representation of the RN-derivative. By means of \(q^U_n(\cdot)\) exponential martingales \((M^{U,\theta}_n)\) are constructed. Their projections to a basis of \(H\) are studied. The analog investigations are made for continuous time, where \(X(t)\) is assumed to be a process with independent increments (Lévy process). As examples mixed hyperbolic processes, mixed Gaussian/inverse Gaussian and mixed variance Gamma processes are considered.
reviewed by: Uwe Küchler
Mathematics Subject Classification ID: 60G51
Mathematics Subject Classification ID: 60J10
Mathematics Subject Classification ID: 60J25
zbMATH DE Number: 1262932
zbMATH Keywords: mixed distributions
zbMATH Keywords: Lévy-processes
zbMATH Keywords: exponential processes
zbMATH Keywords: martingales
author: Bronius Grigelionis
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1262932