Expected utility approximation and portfolio optimisation

Classical portfolio selection problems that optimise expected utility can usually not be solved in closed form. It is natural to approximate the utility function, and we investigate the accuracy of this approximation when using Taylor polynomials. In the important case of a Merton market and power utility we show analytically that increasing the order of the polynomial does not necessarily improve the approximation of the expected utility. The proofs use methods from the theory of parabolic second-order partial differential equations. All results are illustrated by numerical examples.     

A perturbation solution for bacterial growth and bioremediation in a porous medium with bio-clogging

We investigate a flow problem of relevance in bioremediation and develop a mathematical model for transport of contamination by groundwater and the spreading, confinement, and remediation of chemical waste. The model is based on the fluid mass and momentum balance equations and simultaneous transport and consumption of the pollutant (hydrocarbon) and nutrient (oxygen). Particular emphasis is placed on the study of processes involving the full coupling of reaction, transport and mechanical effects. Dimensional analysis and asymptotic reduction are used to simplify the governing equations, which are then solved numerically.     

Wavelet expansions and asymptotic behavior of distributions

We develop a distribution wavelet expansion theory for the space of highly time-frequency localized test functions over the real line S₀(R)⊂S(R) and its dual space , namely, the quotient of the space of tempered distributions modulo polynomials. We prove that the wavelet expansions of tempered distributions converge in . A characterization of boundedness and convergence in is obtained in terms of wavelet coefficients. Our results are then applied to study local and non-local asymptotic properties of Schwartz distributions via wavelet expansions. We provide Abelian and Tauberian type results relating the asymptotic behavior of tempered distributions with the asymptotics of wavelet coefficients.     

Analysis of expanded mixed methods for fourth-order elliptic problems

The recently proposed expanded mixed formulation for numerical solution of second-order elliptic problems is here extended to fourth-order elliptic problems. This expanded formulation for the differential problems under consideration differs from the classical formulation in that three variables are treated, i.e., the displacement, the stress, and the moment tensors. It works for the case where the coefficient of the differential equations is small and does not need to be inverted, or for the case in which the stress tensor of the equations does not need to be symmetric. Based on this new formulation, various mixed finite elements for fourth-order problems are considered; error estimates of quasi-optimal or optimal order depending upon the mixed elements are derived. Implementation techniques for solving the linear system arising from these expanded mixed methods are discussed, and numerical results are presented. © 1997 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 13: 483–503, 1997     

When is the pseudo-best estimator blue?

Summary Linear unbiased estimation of the mean of a random variable in Hilbert space is treated in the typical case where the mean belongs to a known subspace. The best linear estimate depends on the underlying covariance operatorB ₀ of the random variable. This operatorB ₀, however, is rarely completely known, so that an auxiliary operatorB is used to compute a “pseudo-best” estimate. It is shown that the best and the pseudo-best estimates coincide, if and only ifB ₀ B ⁻¹ leavesM invariant. Applications to linear regression are to be found in the references.     

The Spearman footrule and a Markov chain property

An equivalent representation of the Spearman footrule is considered and a characterization in terms of a Markov chain is established. A martingale approach is thereby incorporated in the study of the asymptotic normality of the statistics.     

Repeated Interaction Quantum Systems: Van Hove Limits and Asymptotic States

We establish the existence of two weak coupling regime effective dynamics for an open quantum system of repeated interactions (vanishing strength and individual interaction duration, respectively). This generalizes known results (Attal and Joye in J. Stat. Phys. 126:1241–1283, 2007) in that the von Neumann algebras describing the system and the chain element may not be of finite type. Then (but now assuming that the small system is of finite type), we prove that both effective dynamics capture the long-term behavior of the system: existence of a unique asymptotic state for them implies the same property for the respective exact dynamics—provided that the perturbation parameter is sufficiently small. The zero-th order term in a power series expansion in the perturbation parameter of such an asymptotic state is given by the asymptotic state of the effective dynamics. We conclude by working out the case in which the small system and the chain element are spins. Dedicated to Mariana Huerta. This work was partially funded by Nucleus Millennium Information and Randomness P04-069-F.     

Censored lifetime learning: Optimal Bayesian age-replacement policies

We consider a sequence of age-replacement problems with a general lifetime distribution parametrized by an a-priori unknown parameter. There is a trade-off: Preventive replacements are censored but cheap, whereas corrective replacements are uncensored but costly observations of the lifetime distribution. We first analyze the optimal policy for a finite sequence and establish some properties. We then propose a myopic Bayesian policy that almost surely learns the unknown parameter and converges to the optimal policy with full knowledge of the parameter.