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 X_t = A_tX_t-1 + B_t, t ∈ Z, where ((A_t, B_t))_t∈Z is an independent and identically distributed bivariate sequence. We derive recursive formulae for the fractional moments E|X₀|^p, p ∈ R. Special attention is given to the case when B_t has an Erlang distribution. We provide various approximations to the moments E|X₀|^p and show their performance in a small numerical study.