On mixed exponential processes and martingales
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.
scientific article; zbMATH DE number 1262932