The maximality principle revisited: on certain optimal stopping problems

关于最优停止中极大性原理适用性的一个例子：涉及扩散过程及其运行最大值

On the applicability of the maximality principle in optimal stopping: an example involving a diffusion and its running maximum

本文构造一个定制的最优停止问题，证明极大性原理在该问题中不适用，从而揭示其适用性的边界条件。

A capped optimal stopping problem for the maximum process

This paper concerns an optimal stopping problem driven by the running maximum of a spectrally negative Levy process X. More precisely, we are interested in capped versions of the American lookback optimal stopping problem, which has its origins in mathematical finance, and provide semi-explicit solutions in terms of scale functions. The optimal stopping boundary is characterised by an ordinary first-order differential equation involving scale functions and, in particular, changes according to the path variation of X. Furthermore, we will link these capped problems to Peskir's maximality princi…

An optimal stopping problem for spectrally negative Markov additive processes

Previous authors have considered optimal stopping problems driven by the running maximum of a spectrally negative Lévy process $X$, as well as of a one-dimensional diffusion. Many of the aforementioned results are either implicitly or explicitly dependent on Peskir's maximality principle. In this article, we are interested in understanding how some of the main ideas from these previous works can be brought into the setting of problems driven by the maximum of a class of Markov additive processes (more precisely Markov modulated Lévy processes). Similarly to previous works in the Lévy setting, …

Pathwise inequalities for local time: Applications to Skorokhod embeddings and optimal stopping

We develop a class of pathwise inequalities of the form $H(B_t)\ge M_t+F(L_t)$, where $B_t$ is Brownian motion, $L_t$ its local time at zero and $M_t$ a local martingale. The concrete nature of the representation makes the inequality useful for a variety of applications. In this work, we use the inequalities to derive constructions and optimality results of Vallois' Skorokhod embeddings. We discuss their financial interpretation in the context of robust pricing and hedging of options written on the local time. In the final part of the paper we use the inequalities to solve a class of optimal s…

The naming game in language dynamics revisited

This article studies a biased version of the naming game in which players located on a connected graph interact through successive conversations to bootstrap a common name for a given object. Initially, all the players use the same word B except for one bilingual individual who also uses word A. Both words are attributed a fitness, which measures how often players speak depending on the words they use and how often each word is pronounced by bilingual individuals. The limiting behavior depends on a single parameter: phi = the ratio of the fitness of word A to the fitness of word B. The main ob…