Week 23.09.2024 – 29.09.2024

The Skorokhod reflection was used in 1961 to create a reflected diffusion on the half-line. Later, it was used for processes with jumps such as reflected Lévy processes. Like a Brownian motion, which is a weak limit of random walks, reflected processes on the half-line serve as weak limits of random walks with switching regimes at zero: one regime away from zero, the other around zero. We develop a general theory of this regime change and prove convergence to a function with generalized reflection. Our results are deterministic and can be applied to a wide class of stochastic processes. Applications include storage processes, heavy traffic limits, diffusion on a half-line with a combination of continuous reflection, jump exit, and a delay at 0.

For a convergent positive series, we study the properties of the set of all possible subsums. It is well known that the aforementioned set, up to homeomorphism, is either a finite union of closed intervals, Cantor set, or M-Cantorval. The last case is quite complex and understudied. Formally, M-Cantorval is a perfect set on the real line, which is the closure of its interior, and the endpoint of any nontrivial component of this set are accumulation points of trivial components. Our focus lies in identifying the necessary conditions for the set of subsums to be a Cantorval and investigating its structure.

The coherent state transform, under various names, appears in many fields of mathematics and physics. It is associated with representations of a group. In this talk we are concerned with the problem of minimizing the entropy of the coherent state transform and we explain how complex analysis can be used to achieve this in certain settings. We discuss various open questions.

In order to assess the financial condition of a pension fund, one needs to take into account the mortality forecast so the longevity risk is considered in a consistent way on future cash flows. Usually, the forecast of mortality rates is performed with national or country population data. Even in the presence of basis risk when applying it for pension funds sub-populations (selected populations), for most of the countries this may not be a meaningful problem. However, for countries with relevant social inequalities and a heterogeneous population, national mortality rates may be quite different and more severe than the ones observed in selected sub-populations. In this paper, we use Gaussian processes in a spatial covariance framework applied to sub-population frameworks such that reference populations are used. The applications are performed with a time series of a Brazilian small pension fund population along with the annual country mortality table and also with the use of a public non-periodic insurance industry mortality table. Our aim is to coherently forecast longevity scenarios for the pension fund population. Joint work with Eduardo F. L. de Melo (FGV) and Michael Ludkovski (UCSB).