Incompleteness of the bond market with L\'evy noise under the physical measure

TL;DR

This paper investigates the incompleteness of a bond market model driven by a Lévy process under the physical measure, providing theoretical insights into stochastic integration and martingale representation.

Contribution

It demonstrates the market is incomplete when the Lévy measure has a density and develops the stochastic integration theory for jump measures under a martingale measure.

Findings

Market is incomplete with Lévy measure having a density

Develops stochastic integration over compensated jump measures

Proves integral representation of local martingales

Abstract

The problem of completeness of the forward rate based bond market model driven by a L\'evy process under the physical measure is examined. The incompleteness of market in the case when the L\'evy 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 is presented and the corresponding integral representation of local martingales is proven.