Representation Theory for Stochastic Integrals with Fractional Integrator Processes

This paper considers large sample approximations to the covariances of a nonstationary fractionally integrated processes with the stationary increments of another such process – possibly, itself. Questions of interest include the relationship between the harmonic representation of these random variables, which we have analysed in a previous paper, and the construction derived from moving average representations in the time domain. The limiting integrals are shown to be expressible in terms of functionals of Itô integrals with respect to two distinct Brownian motions. They have an unexpectedly complex structure but possess the required characteristics, in particular an integration by parts formula. The advantages of our approach over the harmonic analysis include the facts that our formulas are valid for the full range of the long memory parameters, and extend to non-Gaussian processes. Corresponding author. School of Business and Economics, University of Exeter, Exeter EX4 4PU, UK. Email: [email protected]