The author studies the problem of completeness of a bond market governed by a Lévy process. It is supposed that the physical measure is not a martingale measure. In this case the transition to the martingale measure is faced with such difficulties as that Lévy processes are not stable under a change of measure and also we need a relevant version of the martingale representation theorem. In this connection, the properties of a Lévy process are discussed that are needed to formulate the martingale decomposition formula and to describe equivalent measures. Then, the construction of a stochastic integral over the compensated jump measure under an equivalent measure and the related martingale representation formula are presented. The definition of admissible strategies is discussed since it depends on the choice of the martingale measure. The main result shows the market incompleteness in the case where the Lévy measure has a density function.
For the entire collection see [Zbl 1314.91009].