L\'evy processes optimal stopping problems and stochastic games

Levy过程可以被认为是一类描述随机运动的粒子的运动或路径的模型，该粒子可能扩散或经历独立的随机跳跃，其量级可以是任意大的或任意小的。Levy过程在其随机结构中内置了几个分布属性，这使得它们在从应用概率领域构建和分析特定主题时作为一种数学工具特别吸引人。最优停止问题是一类数学问题，其中玩家可以停止随机移动的过程，例如Levy过程，以便在停止时获得与随机过程的某个预先指定的函数相等的奖金。一个基本的问题是根据一些优化准则建立一个最优的停止策略。随机博弈是这一主题的变种，其中两个参与者可以停止一个随机移动的过程。他们行为的结果是，无论谁先停止，博弈者1将在停止时获得一个预先指定的随机过程函数，该函数将由博弈者2支付。所使用的预先指定的函数仅取决于谁先停止。这里的一个基本问题是根据合理的优化标准为两个玩家建立停止策略。通常情况下，博弈者1会想要尽可能多地获得财富，而博弈者2会想要尽可能地减少他们的债务的价值。这个项目讨论了当潜在的随机移动过程是Levy过程时，在这类问题中出现的一些数学现象。在现有的文献中，对这类问题的明确解决方案知之甚少。许多数学困难是由于利维过程沿着其轨迹跳跃的方式而产生的。然而，现在已经有了一个足够成熟的Levy过程理论来考察它在这种情况下的应用。在许多情况下，可以将金融市场上奇异期权的定价表示为最优停止问题或随机博弈的解。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，目前的提议非常及时，将在金融数学领域产生直接影响。

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.