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We provide an overview of how the law of large numbers breaks down when pricing life-contingent claims under stochastic as opposed to deterministic mortality (probability, hazard) rates. In a stylized situation, we derive the limiting per-policy risk and show that it goes to a non-zero constant. This is in contrast to the classical situation when the underlying mortality decrements are known with certainty, per policy risk goes to zero. We decompose the standard deviation per policy into systematic and non-systematic components, akin to the analysis of individual stock (equity) risk in a Markowitz portfolio framework. Finally, we draw upon the financial analogy of the Sharpe Ratio to develop a premium pricing methodology under aggregate mortality risk. Copyright The Journal of Risk and Insurance, 2006.

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A result of Milnor [1] states that the infimum of the total curvature of a tame knot K is given by 2πμ(K), where μ(K) is the crookedness of the knot K. It is also known that μ(K)=b(K), where b(K) is the bridge index of K [2]. The situation appears to be quite different for knots realised as polygons in the cubic lattice. We study the total curvature of lattice knots by developing algebraic techniques to estimate minimal curvature in the cubic lattice. We perform simulations to estimte the minimal curvature of lattice knots, and conclude that the situation is very different than for tame knots in ℛ3.

How many edges are necessary and sufficient to construct a knot of type K in the cubic lattice? Define the minimal edge number of a knot to be this number of edges. To what extend does the minimal edge number measure the complexity of a knot? What is the behaviour of the minimal edge number under the connected sum of knots, and what is its limiting behaviour? We consider these questions and show that the minimal edge number may be computed using simulated annealing.

The supermodular order on multivariate distributions has many applications in financial and actuarial mathematics. In the particular case of finite, discrete distributions, we generalize the order to distributions on finite lattices. In this setting, we focus on the generating cone of supermodular functions because the extreme rays of that cone (modulo the modular functions) can be used as test functions to determine whether two random variables are ordered under the supermodular order. We completely determine the extreme supermodular functions in some special cases. (2000): 06B99 (Primary), 52A20 (Secondary). Mathematics Subject Classification

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