Prediction of Lévy-driven CARMA processes

Abstract

The conditional expectations, E(Y(h)|Y(u),−∞<u≤0) and E(Y(h)|Y(u),−M≤u≤0) with h>0 and 0<M<∞ are determined for a continuous-time ARMA (CARMA) process (Y(t))t∈R driven by a Lévy process L with E|L(1)|<∞. If E(L(1)2)<∞ these are the minimum mean-squared error predictors of Y(h) given (Y(t))t≤0 and (Y(t))−M≤t≤0 respectively. Conditions are also established under which the sample-path of L can be recovered from that of Y, both when Y is causal and strictly stationary and (without these assumptions) when L is a pure-jump Lévy process. When E(L(1)2)<∞ and Y is causal and strictly stationary the best linear predictors P(Y(h)|Y(u),u≤0) and P(Y(h)|Y(−nΔ),n∈N) are also determined, the latter yielding a simple algorithm for determining the parameters of the ARMA process obtained by sampling the CARMA process at regular intervals.