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

申请人的研究领域是概率论。他的研究计划将涉及三个截然不同的主题。******应用最多的主题是精算金融，即从退休收入产品的优化设计和管理中产生的数学问题。该研究计划包括该领域的两个项目。一个是关于Tontines的行为和设计。当死亡率不确定或随机时，这些是年金的替代方案。它们对冲了个人的特殊长寿风险（他们比其他人活得更长），但让他们暴露在系统性长寿风险（整个人口比预期活得更长）之下。因此，提供通年金应该比年金更便宜，风险更小，人们有兴趣了解购买者的成本与风险之间的权衡，设计此类产品的最佳方式，以及影响它们如何使个人受益的因素。该领域的第二个项目将研究个人在获得有关其生物年龄（可能与实际年龄不同）的信息后，应该如何从退休储蓄中消费。基因测试将很快使这类信息广泛可用，因此探索它对退休计划的影响（以及它对年金定价和风险管理的影响）是很重要的。******一个完全独立的主题是随机环境下的随机行走研究。这适用于研究通过无序系统的随机运动的一般领域（例如，水通过含水层的渗透）。这一领域的经典工作假设了椭圆性或均匀椭圆性，即步行者总是可以向任何方向移动。最近，人们对放宽这一条件，禁止某些（随机变化的）方向的模型很感兴趣。这导致了渗透问题，以及与以前的工作具有不同特征的障碍或陷阱。在维度2中，我们想要展示平衡但不对称模型的递归性。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）涉及超布朗运动的x -调和函数的行为和性质。超过程是一类被广泛研究的无限维随机过程，它在欧几里德空间上的概率测度集合中取值。它们产生的一种方式是通过种群遗传学模型的限制。x调和函数允许人们调整描述随机过程的规律（度量的鞅变化），并研究新信息如何导致这些规律被修改（调节过程）。关于这些函数的理论是不完整的，而且很难理解。例如，在这种情况下，有一个自然出现的循环，我们对它的存在或唯一性知之甚少。

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.