L\'evy processes optimal stopping problems and stochastic games

利维过程可以被认为是一类描述随机运动粒子的运动或路径的模型，这些粒子可以扩散或经历独立的随机跳跃，其数量级可以任意大或任意小。Levy过程在其随机结构中内置了几个分布特性，这使得它们在构建和分析应用概率领域内的特定主题时，作为数学工具特别有吸引力。其中一个主题构成了本提案的重点；最优停止和随机博弈。最优停止问题是一类数学问题，其中玩家可能会停止随机移动过程（如Levy过程），以便在停止时获得与随机过程的某些预先指定函数相等的奖金。一个基本问题是根据一些优化准则建立最优停车策略。随机游戏是这一主题的变体，其中两个玩家可以停止随机移动过程。他们行动的结果是，无论谁先停下来，参与人1都将在停下来时获得随机过程中预先指定的函数，该函数将由参与人2支付。所使用的预先指定的功能仅取决于谁先停止。这里的一个基本问题是根据合理的优化标准为双方玩家建立停止策略。通常情况下，玩家1希望获得尽可能多的财富，而玩家2则希望尽可能减少自己的义务价值。这个项目处理了当潜在的随机移动过程是一个Levy过程时出现在这类问题中的一些数学现象。在现有的文献中，对这些问题的明确解决方案知之甚少。由于列维过程沿其轨迹跳跃的方式，产生了许多数学上的困难。尽管如此，现在已经有了一个足够完善的列维过程理论，以便研究它在这种情况下的应用。在许多情况下，可以将金融市场中奇异期权的定价表示为最优停止问题或随机博弈的解决方案。由于最近倾向于使用Levy过程作为市场模型中随机性的潜在来源，因此当前的建议非常及时，并且将在金融数学领域引起直接兴趣。

Levy processes may be thought of as a class of models that describe the motion or path of a randomly moving particle which may diffuse or undergo independent random jumps whose order of magnitude may be both arbitrarily large or arbitrarily small. Levy processes have several distributional properties built in to their random structure that make them particularly attractive to work with as a mathematical tool when building and analyzing certain themes from within the field of applied probability. One such theme forms the focus of this proposal; optimal stopping and stochastic games.Optimal stopping problems are a class of mathematical problems in which a player may stop a randomly moving process, such as a Levy process, in order to claim a prize equal in value to some prespecified function of the random process at the time of stopping. A fundamental problem is to establish an optmimal stopping strategy according to some optimization criteria.Stochastic games are a variant on this theme in which two players may stop a randomly moving process. The consequence of their actions is that, whoever stops first, player 1 will receive a prespecified function of the random process at the time of stopping which is to be paid for by player 2. The prespecified function used depends only on who has stopped first. A fundamental problem here is to establish stopping strategies for both players according to sensible optimization criteria. Typically player 1 will want to gain as much wealth as possible whilst player 2 will want to reduce the value of their obligation as much as possible.This project deals with a number of mathematical phenomena which appear in such problems when the underlying randomly moving process is a Levy process. There is very little known about explicit solutions to such problems in the existing literature. Many mathematical difficulties arise becasue of the way in which Levy processes jump along their trajectory. None the less there is now a sufficiently well developed theory of Levy processes in order to look at its application in this context.In many cases, it is possible to express the pricing of exotic options in financial markets as the solution to either an optimal stopping problem or a stochastic game. With the recent preference for the use of Levy processes as an underlying source of randomness in market models, the current proposal is very timely and will be of direct interest within the field of financial mathematics.