Detecting and estimating intensity of jumps for discretely observed ARMAD(1,1) processes

We consider n equidistributed random functions, defined on [0,1], and admitting fixed or random jumps, the context being D[0,1]-valued ARMA(1, 1) processes. We begin with properties of ARMAD(1,1) processes. Next, different scenarios are considered: fixed instants with a given but unknown probability of jumps (the deterministic case), random instants with ordered intensities (the random case), and random instants with non ordered intensities (the completely random case). By using discrete data and for each scenario, we identify the instants of jumps, whose number is either random or fixed, and then estimate their intensity.