Perpetual integral functionals of multidimensional stochastic processes

Abstract

The paper is devoted to the existence of perpetual integral functionals for several classes of d-dimensional of stochastic processes . The method is very simple: we establish the conditions supplying that these functionals have a finite expectation. Examples of these classes include d-dimensional fractional Brownian motion having coordinates with the same Hurst index H, for which existence is established under the assumption . In particular, perpetual integral functionals exist for d-dimensional Brownian motion with d>2, compound Poisson process, Markov processes admitting densities of transitional probabilities. In the case of Brownian motion and fractional Brownian motion we establish that the perpetual integral functionals are not a constant a.s. if .