Bond Market Completeness and Attainable Contingent Claims

A general class, introduced in [Ekeland et al. 2003], of continuous time bond markets driven by a standard cylindrical Brownian motion $\text{wienerq}$ in ${\mathrm{\ell }}^2,$ is considered. We prove that there always exist non-hedgeable random variables in the space $\text{derprod}0={\cap }_{p\ge 1}{L}^p$ and that $\text{derprod}0$ has a dense subset of attainable elements, if the volatility operator is non-degenerated a.e. Such results were proved in [Bj"ork et al. 1997] in the case of a bond market driven by finite dimensional B.m. and marked point processes. We define certain smaller spaces $\text{derprod}s,$ $s>0$ of European contingent claims, by requiring that the integrand in the martingale representation, with respect to $\text{wienerq}$, takes values in weighted ${\mathrm{\ell }}^2$ spaces ${\mathrm{\ell }}^{s,2},$ with a power weight of degree $s\mathrm{.}$ For all $s > 0,$thespace$\derprod{}{s}$isdensein$\derprod{}{0}$ and is independent of the particular bond price and volatility operator processes. A simple condition in terms of ${\mathrm{\ell }}^{s,2}$ norms is given on the volatility operator processes, which implies if satisfied, that every element in $\text{derprod}s$ 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 ${\mathrm{\ell }}^2$-valued market price of risk process has certain Malliavin differentiability properties.