Fractional moments of solutions to stochastic recurrence equations

Abstract

In this paper we study the fractional moments of the stationary solution to the stochastic recurrence equation Xt = AtXt-1 + Bt, t ∈ ℤ, where ((At, Bt)) t∈ℤ is an independent and identically distributed bivariate sequence. We derive recursive formulae for the fractional moments E|X 0|p, p∈ℝ. Special attention is given to the case when Bt has an Erlang distribution. We provide various approximations to the moments E|X0|p and show their performance in a small numerical study. © Applied Probability Trust 2013.