Bond market completeness and attainable contingent claims

A general class, introduced in [5], of continuous time bond markets driven by a standard cylindrical Brownian motion W̄ in l2, is considered. We prove that there always exist non-hedgable random variables in the space D0 = ∩p≥1L and that D0 has a dense subset of attainable elements, if the volatility operator is non-degenerated a.e. Such results were proved in [1] and [2] in the case of a bond market driven by finite dimensional B.m. and marked point processes. We define certain smaller spaces Ds, s > 0 of European contingent claims, by requiring that the integrand in the martingale representation, with respect to W̄ , takes values in weighted l2 spaces ls,2, with a power weight of order s. The spaces Ds, s ≥ 0 are dense in D0 and are independent of the particular bond price and volatility operator processes. A simple condition in terms of ls,2 norms is given on the volatility operator processes, which implies if satisfied, that every element in Ds is attainable. In this context a related problem of optimal portfolios of zero coupon bonds is solved for general utility functions and volatility operator processes, provided that the l2-valued market price of risk process has certain Malliavin differentiability properties.