Completion of a Lévy market by power-jump assets
The authors deal with the so-called (geometric) Lévy market model. Under this model the stock price process is modelled by a stochastic differential equation driven by a general Lévy process. Except for the geometric Brownian and the geometric Poissonian model, there are many equivalent martingale measures for the above-described general geometric Lévy market models, and such markets are incomplete: contingent claims cannot in general be hedged by a self-financing portfolio. The authors propose to enlarge the market by a series of very special assets related to the power-jump processes of the underlying Lévy process. A martingale representation theorem in terms of these (orthonormalized) power-jump processes leads to market completeness. In the second part it is assumed that the market is already enlarged by these so-called power-jump assets. The problem is whether the market is arbitrage free and complete. Existence of an equivalent martingale measure making not only a discounted stock price a martingale, but also all discounted prices of the newly introduced assets is studied. This problem is related to the moment problem.
scientific article; zbMATH DE number 2169556