L\'evy processes optimal stopping problems and stochastic games

Levy 处理最优停止问题和随机博弈

• 批准号: EP/D045460/1
• 负责人: Andreas Kyprianou
• 金额: $ 18.48万
• 依托单位: University of Bath
• 依托单位国家: 英国
• 项目类别: Research Grant
• 财政年份: 2007
• 资助国家: 英国
• 起止时间: 2007 至 无数据
• 项目状态: 已结题
• 来源: https://gtr.ukri.org/projects?ref=EP%2FD045460%2F1
• 关键词: evy; processes; optimal; stopping; problems

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.

列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，其数量级可以是任意大或任意小。利维过程有几个分布特性内置于其随机结构中，这使得它们在构建和分析应用概率领域的某些主题时作为数学工具特别有吸引力。最优停止问题（optimal stopping problem）是一类数学问题，在这类问题中，一个参与者可以停止一个随机运动的过程，例如一个Levy过程，以获得一个与停止时随机过程的某个预先指定的函数相等的奖励。一个基本的问题是建立一个最优的停止策略，根据一些优化准则。随机游戏是一个变种，在这个主题中，两个球员可以停止一个随机移动的过程。他们行动的结果是，无论谁先停止，参与人1将在停止时收到一个随机过程的预先指定的函数，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，玩家1会想获得尽可能多的财富，而玩家2会想尽可能减少他们的债务价值。这个项目涉及的数学现象出现在这样的问题时，潜在的随机移动过程是一个利维过程。在现有的文献中，很少有人知道这类问题的显式解。许多数学上的困难是由于列维过程沿其轨迹沿着跳跃的方式而产生的。尽管如此，现在已经有一个充分发展的Levy过程理论，以便研究它在这方面的应用，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。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，目前的建议非常及时，将在金融数学领域内直接感兴趣。

项目成果

Meromorphic Lévy processes and their fluctuation identities

亚形 Lévy 过程及其涨落特性

• DOI: 10.1214/11-aap787
• 发表时间: 2012
• 期刊: The Annals of Applied Probability
• 作者: Kuznetsov A

The Shepp-Shiryaev Stochastic Game Driven by a Spectrally Negative Lévy Process

由谱负 Lévy 过程驱动的 Shepp-Shiryaev 随机博弈

• DOI: 10.1137/s0040585x97983778
• 发表时间: 2009
• 期刊: Theory of Probability & Its Applications
• 影响因子: 0.6
• 作者: Baurdoux E

Explicit identities for Lévy processes associated to symmetric stable processes

与对称稳定过程相关的 Lévy 过程的显式恒等式

• DOI: 10.3150/10-bej275
• 发表时间: 2011
• 期刊: Bernoulli
• 影响因子: 1.5
• 作者: Caballero M

Fluctuation theory and exit systems for positive self-similar Markov processes

正自相似马尔可夫过程的波动理论和退出系统

• DOI: 10.48550/arxiv.0812.2506
• 发表时间: 2008
• 作者: Chaumont L

On the Lamperti stable processes

关于 Lamperti 稳定过程

• DOI: 10.48550/arxiv.0802.0851
• 发表时间: 2008
• 作者: Caballero M