A refined and asymptotic analysis of optimal stopping problems of Bruss and Weber

The classical secretary problem has been generalized over the years into several directions. In this paper we confine our interest to those generalizations which have to do with the more general problem of stopping on a last observation of a specific kind. We follow Dendievel, (where a bibliography can be found) who studies several types of such problems, mainly initiated by Bruss and Weber. Whether in discrete time or continuous time, whether all parameters are known or must be sequentially estimated, we shall call such problems simply "Bruss-Weber problems". Our contribution in the present paper is a refined analysis of several problems in this class and a study of the asymptotic behaviour of solutions. The problems we consider center around the following model. Let $X_1,X_2,\ldots,X_n$ be a sequence of independent random variables which can take three values: $\{+1,-1,0\}.$ Let p:=\P(X_i=1), p':=\P(X_i=-1), \qt:=\P(X_i=0), p\geq p', where p+p'+\qt=1. The goal is to maximize the probability of stopping on a value $+1$ or $-1$ appearing for the last time in the sequence. Following a suggestion by Bruss, we have also analyzed an x-strategy with incomplete information: the cases $p$ known, $n$ unknown, then $n$ known, $p$ unknown and finally $n,p$ unknown are considered. We also present simulations of the corresponding complete selection algorithm.Comment: 22 pages, 19 figure

Markov Decision Problems Where Means Bound Variances

We identify a rich class of finite-horizon Markov decision problems (MDPs) for which the variance of the optimal total reward can be bounded by a simple linear function of its expected value. The class is characterized by three natural properties: reward nonnegativity and boundedness, existence of a do-nothing action, and optimal action monotonicity. These properties are commonly present and typically easy to check. Implications of the class properties and of the variance bound are illustrated by examples of MDPs from operations research, operations management, financial engineering, and combinatorial optimization

Essays in Problems of Optimal Sequential Decisions

In this dissertation, we study several Markovian problems of optimal sequential decisions by focusing on research questions that are driven by probabilistic and operations-management considerations. Our probabilistic interest is in understanding the distribution of the total reward that one obtains when implementing a policy that maximizes its expected value. With this respect, we study the sequential selection of unimodal and alternating subsequences from a random sample, and we prove accurate bounds for the expected values and exact asymptotics. In the unimodal problem, we also note that the variance of the optimal total reward can be bounded in terms of its expected value. This fact then motivates a much broader analysis that characterizes a class of Markov decision problems that share this important property. In the alternating subsequence problem, we also outline how one could be able to prove a Central Limit Theorem for the number of alternating selections in a finite random sample, as the size of the sample grows to infinity. Our operations-management interest is in studying the interaction of on-the-job learning and learning-by-doing in a workforce-related problem. Specifically, we study the sequential hiring and retention of heterogeneous workers who learn over time. We model the hiring and retention problem as a Bayesian infinite-armed bandit, and we characterize the optimal policy in detail. Through an extensive set of numerical examples, we gain insights into the managerial nature of the problem, and we demonstrate that the value of active monitoring and screening of employees can be substantial

Stochastic Optimisation Problems of Online Selection under Constraints.

PhD ThesesThis thesis deals with several closely related, but subtly di erent problems in the area of sequential stochastic optimisation. A joint property they share is the online constraint that is imposed on the decision-maker: once she observes an element, the decision whether to accept or reject it should be made immediately, without an option to recall the element in future. Observations in these problems are random variables, which take values in either R or in Rd, following known reasonably well-behaving continuous distributions. The stochastic nature of observations and the online condition shape the optimal selection policy. Furthermore, the latter indeed depends on the latest information and is updated at every step. The optimal policies may not be easily described. Even for a small number of steps, solving the optimality recursion may be computationally demanding. However, a detailed investigation yields a range of easily-constructible suboptimal policies that asymptotically perform as well as the optimal one. We aim to describe both optimal and suboptimal policies and study properties of the random processes that arise naturally in these problems. Speci cally, in this thesis we focus on the sequential selection of the longest increasing subsequence in discrete and continuous time introduced by Samuels and Steele [55], the quickest sequential selection of the increasing subsequence of a xed size recently studied by Arlotto et al. [3], and the sequential selection under a sum constraint introduced by Co man et al. [26]

On Dynamic Coherent and Convex Risk Measures : Risk Optimal Behavior and Information Gains

We consider tangible economic problems for agents assessing risk by virtue of dynamic coherent and convex risk measures or, equivalently, utility in terms of dynamic multiple priors and dynamic variational preferences in an uncertain environment. Solutions to the Best-Choice problem for a risky number of applicants are well-known. In Chapter 2, we set up a model with an ambiguous number of applicants when the agent assess utility with multiple prior preferences. We achieve a solution by virtue of multiple prior Snell envelopes for a model based on so called assessments. The main result enhances us with conditions for the ambiguous problem to possess finitely many stopping islands. In Chapter 3 we consider general optimal stopping problems for an agent assessing utility by virtue of dynamic variational preferences. Introducing variational supermartingales and an accompanying theory, we obtain optimal solutions for the stopping problem and a minimax result. To illustrate, we consider prominent examples: dynamic entropic risk measures and a dynamic version of generalized average value at risk. In Chapter 4, we tackle the problem how anticipation of risk in an uncertain environment changes when information is gathered in course of time. A constructive approach by virtue of the minimal penalty function for dynamic convex risk measures reveals time-consistency problems. Taking the robust representation of dynamic convex risk measures as given, we show that all uncertainty is revealed in the limit, i.e. agents behave as expected utility maximizers given the true underlying distribution. This result is a generalization of the fundamental Blackwell-Dubins theorem showing coherent as well as convex risk measures to merge in the long run

Errata and Addenda to Mathematical Constants

We humbly and briefly offer corrections and supplements to Mathematical Constants (2003) and Mathematical Constants II (2019), both published by Cambridge University Press. Comments are always welcome.Comment: 162 page

Further developments on theoretical and computational rheology

Tese financiada pela FCT - Fundação para a Ciência e a Tecnologia, Ciência.Inovação2010, POPH, União Europeia FEDERTese de doutoramento. Engenharia Química e Biológica. Faculdade de Engenharia. Universidade do Porto. 201