Abstract: The detection of change-points in heterogeneous sequences is a statistical challenge with applications across a wide variety of fields, such that fraud and computer intrusion detection, change of risk structure for policy insurance, in bioinformatics for detecting Copy Number Variation(CNV) in DNA, and the uncertainty of change-point location.

In experienced mortality, our first motivation, deaths can be observed sequentially, but detecting change as soon as possible allows us to update mortality assumptions. Given the heterogeneity of the populations, only the ratio of the change may be assumed deterministic, when the reference hazard rate is assumed to be random. Given this double uncertainty, the criterium must be robust with respect of the uncertainty of the hazard rate (intensity) process. So, we adopt the robust Lorden criterion (1971), penalizing the detection delay via its worst-case value, and the false alarm constraint, in place of the classical Bayesian point of view. In the Wiener case with proportional deterministic drift change, such a problem was solved using the so-called cumulative sums (cusum) strategy by many authors, in particular the Russian school (Shiryaev (1963,...,2009)), Peskir, Moustakides (2004) and many others. Even if our setting concerns doubly stochastic (deaths) point process, the strategy is still based on the cumulative sums (cusum) process. Both criteria are invariant by time rescaling, which minimizes the impact of stochastic intensity.

We derive the exact optimality of the cusum stopping rule by using finite variation calculus and elementary martingale properties to characterize the performance functions of the cusum stopping rule. The main difficulty is to obtain a "true" non-asymptotic result, which was proved by Moustakides (2008) in the decreasing case only. From a thin identity on the performance functions, the conjecture is proved. Numerical applications are provided. In the last part, we suggest different directions to apply this result to longevity risk of pension funds, but before the COVID-19 pandemic. [Joint work with S. Loisel and Y. Sahli.]

Abstract: We consider optimal control problems where information about the controlled system is obtained in a discontinuous way. A prototypical example is that of a system driven by a compound Poisson process whose jumps may be addressed by the controller right away only if they are large enough - smaller jumps can be reacted to only after they have hit the system. We show how Meyer σ-fields allow one to most flexibly model such information flows. For the singular control problem of irreversible investment in our prototypical example described before, we compute an explicit solution from the solution to a stochastic representation problem. The general theory for this problem relies on classic optimal stopping results due to El Karoui (1981) who already considered Meyer σ-fields in that context. Our results also provide an explicitly solution to such a Meyer-optimal stopping problem. [This talk is based on joint work with David Besslich.]

Abstract: I will discuss the Machine Learning for Optimal Stopping Problems (mlOSP) computational template and the associated R package. mlOSP presents a unified numerical implementation of Regression Monte Carlo (RMC) approaches to optimal stopping and includes multiple novel RMC variants. These include new options for constructing the simulation designs for training the regressors, as well as new tie-ins of machine learning regression modules. Moreover, the open-source R-based platform allows for full reproducibility and benchmarking of extant algorithms through a persistent catalogue of case studies. I will demonstrate basic use of the package and highlight some of the associated features and benchmarking findings. Extensions to multiple-stopping and impulse-control problems will be mentioned time permitting. The talk is based on the preprint and the GitHub repository.

Abstract: We consider the Lévy model of the perpetual American call and put options with a negative discount rate under Poisson observations. Similar to the continuous observation case as in De Donno et al. (2020), the stopping region that characterizes the optimal stopping time is either a half-line or an interval. The objective of this paper is to obtain explicit expressions of the stopping and continuation regions and the value function, focusing on spectrally positive and negative cases. To this end, we compute the identities related to the first Poisson arrival time to an interval via the scale function and then apply those identities to the computation of the optimal strategies. We also discuss the convergence of the optimal solutions to those in the continuous observation case as the rate of observation increases to infinity. Numerical experiments are also provided. [Joint with Zbigniew Palmowski and José Luis Pérez.]