The problem of completeness of the forward rate based bond market model driven by a Lévy process under the physical measure is examined. The incompleteness of market in the case when the Lévy measure has a density function is shown. The required elements of the theory of stochastic integration over the compensated jump measure under a martingale measure are presented and the corresponding integral representation of local martingales is proven.

Let {Snbe a random walk in the domain of attraction of a stable law $𝒴$, i.e. there exists a sequence of positive real numbers ( an) such that Sn/anconverges in law to $𝒴$. Our main result is that the rescaled process (S⌊nt⌋/an, t≥0), when conditioned to stay positive, converges in law (in the functional sense) towards the corresponding stable Lévy process conditioned to stay positive. Under some additional assumptions, we also prove a related invariance principle for the random walk killed at its first...