Bond market completeness and attainable contingent claims

Abstract

A general class, introduced in [7], of continuous time bond markets driven by a standard cylindrical Brownian motion $\bar{W}$ in $\ell^{2}$ is considered. We prove that there always exist non-hedgeable random variables in the space $\textsf{D}_{0} = \cap_{p \geq 1}L^{p}$ and that $\textsf{D}_{0}$ has a dense subset of attainable elements, if the volatility operator is non-degenerate a.e. Such results were proved in [1] and [2] in the case of a bond market driven by finite dimensional Brownian motions and marked point processes. We define certain smaller spaces $\textsf{D}_{s}$ , s > 0, of European contingent claims by requiring that the integrand in the martingale representation with respect to $\bar{W}$ takes values in weighted $\ell^{2}$ spaces $\ell^{s,2}$ , with a power weight of degree s. For all s > 0, the space $\textsf{D}_{s}$ is dense in $\textsf{D}_{0}$ and is independent of the particular bond price and volatility operator processes. A simple condition in terms of $\ell^{s,2}$ norms is given on the volatility operator processes, which implies if satisfied that every element in $\textsf{D}_{s}$ 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 $\ell^{2}$ -valued market price of risk process has certain Malliavin differentiability properties.