Completion of a Lévy market by power-jump assets

We work under a geometric Lévy market model: the stock price process is modelled by a SDE driven by a general Lévy process (taking into account jumps). Except for the geometric Brownian model and the geometric Poissonian model, the above described general geometric Lévy market models are incomplete models and there are many equivalent martingale measures. In this paper we suggest to enlarge the market by a series of very special assets (power-jump assets) related to the power-jump processes of the underlying Lévy process suitable compensated. By doing this we show that the market can be completed. The very particular choice of the compensators needed to make these processes tradable is very delicate. For some special cases, we will derive criteria to ensure their introduction in the market will not lead to arbitrage opportunities. The question in general is related to the moment problem.