c © Springer-Verlag 1997 Towards a general theory of bond markets?

To the memory of our friend and colleague Oliviero Lessi. Abstract. The main purpose of the paper is to provide a mathematical back-ground for the theory of bond markets similar to that available for stock markets. We suggest two constructions of stochastic integrals with respect to processes taking values in a space of continuous functions. Such integrals are used to define the evolution of the value of a portfolio of bonds corresponding to a trad-ing strategy which is a measure-valued predictable process. The existence of an equivalent martingale measure is discussed and HJM-type conditions are derived for a jump-diffusion model. The question of market completeness is considered as a problem of the range of a certain integral operator. We introduce a concept of approximate market completeness and show that a market is approximately complete iff an equivalent martingale measure is unique. Key words: Bond market, term structure of interest rates, stochastic integral, Ba-nach space-valued integrators, measure-valued portfolio, jump-diffusion model, martingale measure, arbitrage, market completeness