An improvement of the Kolmogorov–Riesz compactness theorem

The purpose of this short note is to provide a new and very short proof of a result by Sudakov (1957), offering an important improvement of the classical result by Kolmogorov–Riesz on compact subsets of Lebesgue spaces.     

Robust equilibrium excess-of-loss reinsurance and CDS investment strategies for a mean–variance insurer with ambiguity aversion

This paper considers the robust equilibrium reinsurance and investment strategies for an ambiguity-averse insurer under a dynamic mean–variance criterion. The insurer is allowed to purchase excess-of-loss reinsurance and invest in a financial market consisting of a risk-free asset and a credit default swap (CDS). Following a game theoretic approach, robust equilibrium strategies and equilibrium value functions for the pre-default case and the post-default case are derived, respectively. For the ambiguity-averse insurer, in general the equilibrium strategies can be characterized by unique solutions to some algebraic equations. For the degenerate case with an ambiguity-neutral insurer, closed-form expressions of equilibrium strategies and equilibrium value functions are obtained. Numerical examples demonstrate that the consideration of model uncertainty and CDS investment improves the insurer’s utility. In this regard, our paper establishes theoretical and numerical support for the importance of ambiguity aversion, credit risk and their interplay in insurance business.     

Optimal investment strategies and risk-sharing arrangements for a hybrid pension plan

A continuous time stochastic model is used to study a hybrid pension plan, where both the contribution and benefit levels are adjusted depending on the performance of the plan, with risk sharing between different generations. The pension fund is invested in a risk-free asset and multiple risky assets. The objective is to seek an optimal investment strategy and optimal risk-sharing arrangements for plan trustees and participants so that this proposed hybrid pension system provides adequate and stable income to retirees while adjusting contributions effectively, as well as keeping its sustainability in the long run. These goals are achieved by minimizing the expected discount disutility of intermediate adjustment for both benefits and contributions and that of terminal wealth in finite time horizon. Using the stochastic optimal control approach, closed-form solutions are derived under quadratic loss function and exponential loss function. Numerical analysis is presented to illustrate the sensitivity of the optimal strategies to parameters of the financial market and how the optimal benefit changes with respect to different risk aversions. Through numerical analysis, we find that the optimal strategies do adjust the contributions and retirement benefits according to fund performance and model objectives so the intergenerational risk sharing seem effectively achieved for this collective hybrid pension plan.     

Explicit estimates of sums related to the Nyman–Beurling criterion for the Riemann Hypothesis

We give an estimate for sums appearing in the Nyman–Beurling criterion for the Riemann Hypothesis. These sums contain the Möbius function and are related to the imaginary part of the Estermann zeta function. The estimate is remarkably sharp in comparison to other sums containing the Möbius function. The bound is smaller than the trivial bound – essentially the number of terms – by a fixed power of that number. The exponent is made explicit. The methods intensively use tools from the theory of continued fractions and from the theory of Fourier series.