Prediction of Lévy-driven CARMA processes

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 purejump 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 (−n∆), 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. JEL Classification: C02, G17