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

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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.