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 \wienerq{}{} in $\ell^{2},$ is considered. We prove that there always exist non-hedgeable random variables in the space \derprod{}{0}=\cap_{p \geq 1}L^{p} and that \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 \derprod{}{s}, $s>0$ of European contingent claims, by requiring that the integrand in the martingale representation, with respect to \wienerq{}{}, takes values in weighted $\ell^{2}$ spaces $\ell^{s,2},$ with a power weight of degree $s.$ For all $s > 0,$ the space \derprod{}{s} is dense in \derprod{}{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 \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 $\ell^{2}$-valued market price of risk process has certain Malliavin differentiability properties.Comment: 27 pages, Revised version to be published in Finance and Stochastic

Equity Allocation and Portfolio Selection in Insurance

A discrete time probabilistic model, for optimal equity allocation and portfolio selection, is formulated so as to apply to (at least) reinsurance. In the context of a company with several portfolios (or subsidiaries), representing both liabilities and assets, it is proved that the model has solutions respecting constraints on ROE's, ruin probabilities and market shares currently in practical use. Solutions define global and optimal risk management strategies of the company. Mathematical existence results and tools, such as the inversion of the linear part of the Euler-Lagrange equations, developed in a preceding paper in the context of a simplified model are essential for the mathematical and numerical construction of solutions of the model.Comment: 24 pages, LaTeX2

Equity Allocation and Portfolio Selection in Insurance: A simplified Portfolio Model

A quadratic discrete time probabilistic model, for optimal portfolio selection in (re-)insurance is studied. For positive values of underwriting levels, the expected value of the accumulated result is optimized, under constraints on its variance and on annual ROE's. Existence of a unique solution is proved and a Lagrangian formalism is given. An effective method for solving the Euler-Lagrange equations is developed. The approximate determination of the multipliers is discussed. This basic model is an important building block for more complete models.Comment: 31 pages, LaTeX2

A theory of bond portfolios

We introduce a bond portfolio management theory based on foundations similar to those of stock portfolio management. A general continuous-time zero-coupon market is considered. The problem of optimal portfolios of zero-coupon bonds is solved for general utility functions, under a condition of no-arbitrage in the zero-coupon market. A mutual fund theorem is proved, in the case of deterministic volatilities. Explicit expressions are given for the optimal solutions for several utility functions.Comment: Published at http://dx.doi.org/10.1214/105051605000000160 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Robust no-free lunch with vanishing risk, a continuum of assets and proportional transaction costs

We propose a continuous time model for financial markets with proportional transactions costs and a continuum of risky assets. This is motivated by bond markets in which the continuum of assets corresponds to the continuum of possible maturities. Our framework is well adapted to the study of no-arbitrage properties and related hedging problems. In particular, we extend the Fundamental Theorem of Asset Pricing of Guasoni, R\'asonyi and L\'epinette (2012) which concentrates on the one dimensional case. Namely, we prove that the Robust No Free Lunch with Vanishing Risk assumption is equivalent to the existence of a Strictly Consistent Price System. Interestingly, the presence of transaction costs allows a natural definition of trading strategies and avoids all the technical and un-natural restrictions due to stochastic integration that appear in bond models without friction. We restrict to the case where exchange rates are continuous in time and leave the general c\`adl\`ag case for further studies.Comment: 41 page

Generalized integrands and bond portfolios: Pitfalls and counter examples

We construct Zero-Coupon Bond markets driven by a cylindrical Brownian motion in which the notion of generalized portfolio has important flaws: There exist bounded smooth random variables with generalized hedging portfolios for which the price of their risky part is $+\infty$ at each time. For these generalized portfolios, sequences of the prices of the risky part of approximating portfolios can be made to converges to any given extended real number in $[-\infty,\infty].$Comment: Published in at http://dx.doi.org/10.1214/10-AAP694 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Simple Non Linear Klein-Gordon Equations in 2 space dimensions, with long range scattering

We establish that solutions, to the most simple NLKG equations in 2 space dimensions with mass resonance, exhibits long range scattering phenomena. Modified wave operators and solutions are constructed for these equations. We also show that the modified wave operators can be chosen such that they linearize the non-linear representation of the Poincar\'e group defined by the NLKG.Comment: 19 pages, LaTeX, To appear in Lett. Math. Phy