Conditioned stochastic differential equations: theory, examples and application to finance

We generalize the notion of Brownian bridge. More precisely, we study a standard Brownian motion for which a certain functional is conditioned to follow a given law. Such processes appear as weak solutions of stochastic differential equations that we call conditioned stochastic differential equations. The link with the theory of initial enlargement of filtration is made and after a general presentation several examples are studied: the conditioning of a standard Brownian motion (and more generally of a Markov diffusion) by its value at a given date, the conditioning of a geometric Brownian motion with negative drift by its quadratic variation and finally the conditioning of a standard Brownian motion by its first hitting time of a given level. As an application, we introduce the notion of weak information on a complete market, and we give a “quantitative” value to this weak information.     

On a conjecture of Dixon and other topics in variable metric methods

A conjecture of Dixon relating to the behaviour of variable metric methods on functions with special symmetry is validated under suitable onditions. The relation between Huang's class and Oren's class is explored. Then the equivalence of Davidon's and Oren and Spedicato's approaches to optimal conditioning is demonstrated.     

The chemistry of the phosphates of barium and tetravalent cations in the 1:1 stoichiometry

The chemistry of phosphates of barium and tetravalent cations [BaM^IV(PO₄)₂] is reviewed. Such phosphates crystallise in the C2/m space group for M^IV=Ti, Zr, Hf, Ge, Sn, and Mo, and in the P2₁/n space group for BaTh(PO₄)₂. The existence of BaM^IV(PO₄)₂ in which M^IV=Pb, Ce, and U is further evaluated. Several aspects, such as phase transitions in the compounds with yavapaiite structure, solid solutions of BaM^IV(PO₄)₂ compounds and practical applications are briefly discussed.     

A pivoted LLL algorithm

Luk and Tracy (2008) [7] developed a matrix interpretation of the LLL algorithm. Building on their work [7], we propose to add pivoting to the algorithm. We prove that our new algorithm always terminates, and we construct a class of ill-conditioned reduced matrices to illustrate the advantages of pivoting.     

Metric regularity of semi-infinite constraint systems

We obtain a formula for the modulus of metric regularity of a mapping defined by a semi-infinite system of equalities and inequalities. Based on this formula, we prove a theorem of Eckart-Young type for such set-valued infinite-dimensional mappings: given a metrically regular mapping F of this kind, the infimum of the norm of a linear function g such that F+g is not metrically regular is equal to the reciprocal to the modulus of regularity of F. The Lyusternik-Graves theorem gives a straightforward extension of these results to nonlinear systems. We also discuss the distance to infeasibility for homogeneous semi-infinite linear inequality systems. Dedicated to R. T. Rockafellar on his 70th Birthday Research partially supported by grants BFM2002-04114-C02 (01-02) from MCYT (Spain) and FEDER (E.U.), GV04B-648 and GRUPOS04/79 from Generalitat Valenciana (Spain), and Bancaja-UMH (Spain).     

Coderivative calculus and metric regularity for constraint and variational systems

This paper provides new developments in generalized differentiation theory of variational analysis with their applications to metric regularity of parameterized constraint and variational systems in finite-dimensional and infinite-dimensional spaces. Our approach to the study of metric regularity for these two major classes of parametric systems is based on appropriate coderivative constructions for set-valued mappings and on extended calculus rules supporting their computation and estimation. The main attention is paid in this paper to the so-called reversed mixed coderivative, which is of crucial importance for efficient pointwise characterizations of metric regularity in the general framework of set-valued mappings between infinite-dimensional spaces. We develop new calculus results for the latter coderivative that allow us to compute it for large classes of parametric constraint and variational systems. On this basis we derive verifiable sufficient conditions, necessary conditions as well as complete characterizations for metric regularity of such systems with computing the corresponding exact bounds of metric regularity constants/moduli. This approach allows us to reveal general settings in which metric regularity fails for major classes of parametric variational systems. Furthermore, the developed coderivative calculus leads us also to establishing new formulas for computing the radius of metric regularity for constraint and variational systems, which characterize the maximal region of preserving metric regularity under linear (and other types of) perturbations and are closely related to conditioning aspects of optimization.     

Poisson-Nernst-Planck Equations for Simulating Biomolecular Diffusion-Reaction Processes I: Finite Element Solutions

In this paper we developed accurate finite element methods for solving 3-D Poisson-Nernst-Planck (PNP) equations with singular permanent charges for electrodiffusion in solvated biomolecular systems. The electrostatic Poisson equation was defined in the biomolecules and in the solvent, while the Nernst-Planck equation was defined only in the solvent. We applied a stable regularization scheme to remove the singular component of the electrostatic potential induced by the permanent charges inside biomolecules, and formulated regular, well-posed PNP equations. An inexact-Newton method was used to solve the coupled nonlinear elliptic equations for the steady problems; while an Adams-Bashforth-Crank-Nicolson method was devised for time integration for the unsteady electrodiffusion. We numerically investigated the conditioning of the stiffness matrices for the finite element approximations of the two formulations of the Nernst-Planck equation, and theoretically proved that the transformed formulation is always associated with an ill-conditioned stiffness matrix. We also studied the electroneutrality of the solution and its relation with the boundary conditions on the molecular surface, and concluded that a large net charge concentration is always present near the molecular surface due to the presence of multiple species of charged particles in the solution. The numerical methods are shown to be accurate and stable by various test problems, and are applicable to real large-scale biophysical electrodiffusion problems.     

Role of pretreatment and conditioning processes on toxicity of lignocellulosic biomass hydrolysates

The Department of Energy’s Office of the Biomass Program has set goals of making ethanol cost competitive by 2012 and replacing 30% of 2004 transportation supply with biofuels by 2030. Both goals require improvements in conversions of cellulosic biomass to sugars as well as improvements in fermentation rates and yields. Current best pretreatment processes are reasonably efficient at making the cellulose/hemicellulose/lignin matrix amenable to enzymatic hydrolysis and fermentation, but they release a number of toxic compounds into the hydrolysate which inhibit the growth and ethanol productivity of fermentation organisms. Conditioning methods designed to reduce the toxicity of hydrolysates are effective, but add to process costs and tend to reduce sugar yields, thus adding significantly to the final cost of production. Reducing the cost of cellulosic ethanol production will likely require enhanced understanding of the source and mode of action of hydrolysate toxic compounds, the means by which some organisms resist the actions of these compounds, and the methodology and mechanisms for conditioning hydrolysate to reduce toxicity. This review will provide an update on the state of knowledge in these areas and can provide insights useful for the crafting of hypotheses for improvements in pretreatment, conditioning, and fermentation organisms.     

Underestimation of total arsenic concentration in groundwater samples determined by hydride generation quartz furnace atomic absorption spectrometry due to sample characteristics

Total arsenic concentrations in groundwater samples determined by hydride generation quartz furnace atomic absorption spectrometry may underestimated due to a loss of quartz cell surface “conditioning.” This loss of quartz cell surface “conditioning” is due to the analysis of many samples that do not contain measurable quantities of the analyte. This process is further accelerated by the use of high concentrations of sodium tetrahydroborate and hydrochloric acid. Analysis of the highest calibration standard in between the samples and the use of low concentrations of sodium tetrahydroborate and hydrochloric acid could minimize the error arising from this source.