Actuarial finance, random walk in random environment, super Brownian motion

精算金融、随机环境中的随机游走、超布朗运动

• 批准号: RGPIN-2017-05706
• 负责人: Salisbury, Thomas
• 金额: $ 2.7万
• 依托单位: York University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2022
• 资助国家: 加拿大
• 起止时间: 2022-01-01 至 2023-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=759116
• 关键词: Actuarial; finance; random; walk; environment

The applicant's area of study is probability theory. His research program will address three quite distinct topics.The most applied topic is in actuarial finance, namely mathematical questions arising from the optimal design and management of retirement income products. The research program includes two projects in this area. One concerns the behaviour and design of Tontines. These are alternatives to annuities, when mortality rates are uncertain or stochastic. They hedge individuals' idiosyncratic longevity risk (the risk that they will live longer than others), but leave them exposed to systematic longevity risk (the risk that the entire population will live longer than anticipated). Tontines should therefore be cheaper and less risky to provide than annuities, and one is interested in understanding the tradeoff of purchasers' cost versus risk, the optimal way to design such products, and the factors that affect how they benefit individuals. A second project in this area will study how individuals should consume from a retirement nest egg, once they have access to information about their biological age (which may differ from their chronological age). Genetic testing will soon make this kind of information widely available, so it is important to explore its consequence for retirement planning (as well as its consequences for the pricing and risk management of annuities). A completely separate topic is the study of random walk in random environment. This fits into the general field of studying random motion through disordered systems (for example, the percolation of water through an aquifer). The classical work in this area assumes ellipticity or uniform ellipticity, ie that the walker can always move in any direction. Recently there has been interest in models where this condition is relaxed, and some (randomly varying) directions are prohibited. This leads to percolation questions, and to barriers or traps that have a different character than in previous work. In dimension 2 one would like to show recurrence for balanced but asymmetric models. In dimension 3, the percolation questions to resolve will involve random surfaces. The third major topic (also completely separate) concerns the behaviour and properties of X-harmonic functions of super Brownian motion. Superprocesses are a widely studied class of infinite-dimensional stochastic processes, taking values in the set of probability measures on Euclidean space. One way they arise is via limits of population genetics models. X-harmonic functions allow one to adjust the laws which describe the stochastic process (a martingale change of measure), and to study how new information causes those laws to be revised (conditioning the process). The theory of such functions is fragmentary and poorly understood. For example, there is a recurrence that arises naturally in this context, for which we know very little about either existence or uniqueness.

申请人的研究领域是概率理论。他的研究计划将解决三个完全不同的主题。最应用的主题是精算融资，即是由于退休收入产品的最佳设计和管理引起的数学问题。该研究计划包括该领域的两个项目。一种涉及龙门的行为和设计。当死亡率不确定或随机性时，这些是年金的替代品。他们对冲个人的特质寿命风险（他们的寿命比其他人更长的风险），但使他们面临系统的寿命风险（整个人群的寿命比预期的更长的风险）。因此，与年金相比，汤丁的提供应更便宜且风险较小，并且有兴趣了解购买者成本与风险的权衡，设计此类产品的最佳方式以及影响其受益个人的因素。该领域的第二个项目将研究个人应如何从退休巢鸡蛋中消耗的，一旦他们获得了有关其生物年龄的信息（可能与年代年龄有所不同）。基因测试将很快使这种信息广泛可用，因此探索其对退休计划的后果非常重要（以及其对年金的定价和风险管理的后果）。一个完全独立的主题是在随机环境中随机步行的研究。这符合通过无序系统研究随机运动的一般领域（例如，通过含水层渗透水）。该区域的经典作品假定椭圆形或均匀的椭圆度，即助行器总是可以朝任何方向移动。最近，人们对这种情况放松的模型引起了兴趣，并且禁止某些（随机变化）方向。这导致了渗透问题，以及与以前的工作不同的障碍或陷阱。在维度2中，一个人希望显示平衡但不对称模型的复发。在维度3中，要解决的渗透问题将涉及随机表面。第三个主要主题（也完全独立）涉及超级布朗运动的X谐波功能的行为和特性。超级过程是一类经过广泛研究的无限二维随机过程，在欧几里得空间的一系列概率度量中进行了值。它们出现的一种方式是通过人口遗传学模型的限制。 X频函数允许人们调整描述随机过程的法律（Martingale的量度变化），并研究新信息如何导致这些法律进行修订（调节过程）。这种功能的理论是零散的，知之甚少。例如，在这种情况下，自然出现了一种复发，我们对此一无所知。