A generalized Schroedinger equation for loop quantum cosmology

A temporally discrete Schroedinger time evolution equation is proposed for isotropic quantum cosmology coupled to a massless scalar source. The approach employs dynamically determined intrinsic time and produces the correct semiclassical limit.Comment: 5 pages, to appear in the Proceedings of the Eleventh Marcel Grossmann Meetin

Conditioning super-Brownian motion on its boundary statistics, and fragmentation

We condition super-Brownian motion on "boundary statistics" of the exit measure $X_D$ from a bounded domain $D$. These are random variables defined on an auxiliary probability space generated by sampling from the exit measure $X_D$. Two particular examples are: conditioning on a Poisson random measure with intensity $\beta X_D$ and conditioning on $X_D$ itself. We find the conditional laws as $h$-transforms of the original SBM law using Dynkin's formulation of $X$-harmonic functions. We give explicit expression for the (extended) $X$-harmonic functions considered. We also obtain explicit constructions of these conditional laws in terms of branching particle systems. For example, we give a fragmentation system description of the law of SBM conditioned on $X_D=\nu$, in terms of a particle system, called the backbone. Each particle in the backbone is labeled by a measure $\tilde{\nu}$, representing its descendants' total contribution to the exit measure. The particle's spatial motion is an $h$-transform of Brownian motion, where $h$ depends on $\tilde{\nu}$. At the particle's death two new particles are born, and $\tilde{\nu}$ is passed to the newborns by fragmentation.Comment: Published in at http://dx.doi.org/10.1214/12-AOP778 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Valuation and hedging of the ruin-contingent life annuity (RCLA)

This paper analyzes a novel type of mortality contingent-claim called a ruin-contingent life annuity (RCLA). This product fuses together a path-dependent equity put option with a "personal longevity" call option. The annuitant's (i.e. long position) payoff from a generic RCLA is \$1 of income per year for life, akin to a defined benefit pension, but deferred until a pre-specified financial diffusion process hits zero. We derive the PDE and relevant boundary conditions satisfied by the RCLA value (i.e. the hedging cost) assuming a complete market where No Arbitrage is possible. We then describe some efficient numerical techniques and provide estimates of a typical RCLA under a variety of realistic parameters. The motivation for studying the RCLA on a stand-alone basis is two-fold. First, it is implicitly embedded in approximately \$1 trillion worth of U.S. variable annuity (VA) policies; which have recently attracted scrutiny from financial analysts and regulators. Second, the U.S. administration - both Treasury and Department of Labor - have been encouraging Defined Contribution (401k) plans to offer stand-alone longevity insurance to participants, and we believe the RCLA would be an ideal and cost effective candidate for that job

Jurassic stratigraphy of the southern two-thirds of North Dakota

Jurassic rocks in the southern two-thirds of North Dakota consist mainly of limestone, shale, anhydrite, and salt of the Piper Formation; shaly limestone, calcareous and non-calcareous shales of the Rierden Formation; shale and sandstone of the Swift Formation; and, at the top, non-marine shale and local sandstones of the Morrison Formation. The units generally are conformable in the deep part of the Williston Basin, and are unconformable near the edge of the basin in eastern North Dakota. The Poe Member of the Piper Formation, however, unconformably overlies the Spearfish Formation in the western part of North Dakota where Dunham salt is absent; where the Dunham is present, the sequence is conformable. Also, the Morrison Formation is unconformably overlain by the Cretaceous Dakota Group except in local areas where the boundary may be gradational. Mechanical-log characteristics or the Jurassic formations, upon which the study was based, are such that units may be correlated with assurance in the subsurface in the western half of the study area. Facies changes and the progressive thinning of the units make correlation more difficult toward the edge of the basin to the east. Piper limestones and Swift sandstones appear to be the most promising for petroleum potential, although to date no oil has been discovered in the Jurassic rocks within the study area. The Poe Member may contain gypsum at the eastern edge of the basin. The Dunham beds are a potential source of salt in western North Dakota