In the problem of optimal investment with utility function defined on (0, ∞), we formulate sufficient conditions for the dual optimizer to be a uniformly integrable martingale. Our key requirement consists of the existence of a martingale measure whose density process satisfies the probabilistic Muckenhoupt (A p ) condition for the power p = 1/(1− a), where a ∈ (0, 1) is a lower bound on the relative risk-aversion of the utility function. We construct a counterexample showing that this (A p ) condition is sharp.

Stability of the utility maximization problem with random endowment and indifference prices is studied for a sequence of financial markets in an incomplete Brownian setting. Our novelty lies in the nonequivalence of markets, in which the volatility of asset prices (as well as the drift) varies. Degeneracies arise from the presence of nonequivalence. In the positive real line utility framework, a counterexample is presented showing that the expected utility maximization problem can be unstable. A positive stability result is proven for utility functions on the entire real line.

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Three novel dibenzo-18-crown-6 aldimines were successfully synthesised and structurally characterised via various spectroscopic methods (1H,13H NMR, FT-IR) and their solution phase lead binding behaviours probed via absorption spectroscopy, the results are supported by Density Functional Theoretical (DFT) modelling. These methods revealed that the asymmetric nature of these compounds is such that at equilibrium the ether cavity adopts an open configuration where the constituent oxygen atoms exhibit a highly negative electrostatic potential; hence, they spontaneously (ΔG~−58 kJ mol−1) interact/bind aqueous lead ions to form stable 2:1 metal–ligand complexes. As indicated by cyclic and square voltammetry studies, all compounds are redox active and polymerise relatively easily onto a platinum surface to form a multi-layered lead Ion-selective Membrane (ISM), the structure of which is confirmed by Scanning Electron Microscopy (SEM) and Electrochemical Impedance Spectroscopy (EIS). These novel Ion-selective Electrodes (ISEs), as characterised by Differential Pulse Anodic Stripping Voltammetry (D PASV), allow selective electrochemical detection and quantification of lead at concentrations as low as 10 ppm, over a range of 15–60 ppm, with only minimal interference from mercury(II) and aluminium(III) ions at a 1:1 analyte-interferent ratio.