Geometric stopping of a random walk and its applications to valuing equity-linked death benefits

We study discrete-time models in which death benefits can depend on a stock price index, the logarithm of which is modeled as a random walk. Examples of such benefit payments include put and call options, barrier options, and lookback options. Because the distribution of the curtate-future-lifetime can be approximated by a linear combination of geometric distributions, it suffices to consider curtate-future-lifetimes with a geometric distribution. In binomial and trinomial tree models, closed-form expressions for the expectations of the discounted benefit payment are obtained for a series of options. They are based on results concerning geometric stopping of a random walk, in particular also on a version of the Wiener-Hopf factorization.

The average inter-crossing number of equilateral random walks and polygons

In this paper, we study the average inter-crossing number between two random walks and two random polygons in the three-dimensional space. The random walks and polygons in this paper are the so-called equilateral random walks and polygons in which each segment of the walk or polygon is of unit length. We show that the mean average inter-crossing number ICN between two equilateral random walks of the same length n is approximately linear in terms of n and we were able to determine the prefactor of the linear term, which is a = (3 In 2)/(8) approximate to 0.2599. In the case of two random polygons of length n, the mean average inter-crossing number ICN is also linear, but the prefactor of the linear term is different from that of the random walks. These approximations apply when the starting points of the random walks and polygons are of a distance p apart and p is small compared to n. We propose a fitting model that would capture the theoretical asymptotic behaviour of the mean average ICN for large values of p. Our simulation result shows that the model in fact works very well for the entire range of p. We also study the mean ICN between two equilateral random walks and polygons of different lengths. An interesting result is that even if one random walk (polygon) has a fixed length, the mean average ICN between the two random walks (polygons) would still approach infinity if the length of the other random walk (polygon) approached infinity. The data provided by our simulations match our theoretical predictions very well.

Predicting optimal lengths of random knots

In a thermally fluctuating long linear polymeric chain in a solution, the ends, from time to time, approach each other. At such an instance, the chain can be regarded as closed and thus will form a knot or rather a virtual knot. Several earlier studies of random knotting demonstrated that simpler knots show a higher occurrence for shorter random walks than do more complex knots. However, up to now there have been no rules that could be used to predict the optimal length of a random walk, i.e. the length for which a given knot reaches its highest occurrence. Using numerical simulations, we show here that a power law accurately describes the relation between the optimal lengths of random walks leading to the formation of different knots and the previously characterized lengths of ideal knots of a corresponding type.

Self-avoiding random walks and Olbers' paradox

In this paper, we prove that a self-avoiding walk of infinite length provides a structure that would resolve Olbers' paradox. That is, if the stars of a universe were distributed like the vertices of an infinite random walk with each segment length of about a parsec, then the night sky could be as dark as actually observed on the Earth. Self-avoiding random walk structure can therefore resolve the Olbers' paradox even in a static universe.

Animal dispersal modelling: handling landscape features and related animal choices

Animal dispersal in a fragmented landscape depends on the complex interaction between landscape structure and animal behavior. To better understand how individuals disperse, it is important to explicitly represent the properties of organisms and the landscape in which they move. A common approach to modelling dispersal includes representing the landscape as a grid of equal sized cells and then simulating individual movement as a correlated random walk. This approach uses a priori scale of resolution, which limits the representation of all landscape features and how different dispersal abilities are modelled. We develop a vector-based landscape model coupled with an object-oriented model for animal dispersal. In this spatially explicit dispersal model, landscape features are defined based on their geographic and thematic properties and dispersal is modelled through consideration of an organism's behavior, movement rules and searching strategies (such as visual cues). We present the model's underlying concepts, its ability to adequately represent landscape features and provide simulation of dispersal according to different dispersal abilities. We demonstrate the potential of the model by simulating two virtual species in a real Swiss landscape. This illustrates the model's ability to simulate complex dispersal processes and provides information about dispersal such as colonization probability and spatial distribution of the organism's path.

Fission yeast Sec3 and Exo70 are transported on actin cables and localize the exocyst complex to cell poles.

The exocyst complex is essential for many exocytic events, by tethering vesicles at the plasma membrane for fusion. In fission yeast, polarized exocytosis for growth relies on the combined action of the exocyst at cell poles and myosin-driven transport along actin cables. We report here the identification of fission yeast Schizosaccharomyces pombe Sec3 protein, which we identified through sequence homology of its PH-like domain. Like other exocyst subunits, sec3 is required for secretion and cell division. Cells deleted for sec3 are only conditionally lethal and can proliferate when osmotically stabilized. Sec3 is redundant with Exo70 for viability and for the localization of other exocyst subunits, suggesting these components act as exocyst tethers at the plasma membrane. Consistently, Sec3 localizes to zones of growth independently of other exocyst subunits but depends on PIP(2) and functional Cdc42. FRAP analysis shows that Sec3, like all other exocyst subunits, localizes to cell poles largely independently of the actin cytoskeleton. However, we show that Sec3, Exo70 and Sec5 are transported by the myosin V Myo52 along actin cables. These data suggest that the exocyst holocomplex, including Sec3 and Exo70, is present on exocytic vesicles, which can reach cell poles by either myosin-driven transport or random walk.

Climate oscillations and species interactions: large-scale congruence but regional differences in the phylogeographic structures of an alpine plant and its monophagous insect

Aim. To predict the fate of alpine interactions involving specialized species, using a monophagous beetle and its host-plant as a case study. Location. The Alps. Methods. We investigated genetic structuring of the herbivorous beetle Oreina gloriosa and its specific host-plant Peucedanum ostruthium. We used genome fingerprinting (in the insect and the plant) and sequence data (in the insect) to compare the distribution of the main gene pools in the two associated species and to estimate divergence time in the insect, a proxy for the temporal origin of the interaction. We quantified the similarity in spatial genetic structures by performing a Procrustes analysis, a tool from the shape theory. Finally, we simulated recolonization of an empty space analogous to the deglaciated Alps just after ice retreat by two lineages from two species showing unbalanced dependence, to examine how timing of the recolonization process, as well as dispersal capacities of associated species, could explain the observed pattern. Results. Contrasting with expectations based on their asymmetrical dependence, patterns in the beetle and plant were congruent at a large scale. Exceptions occurred at a regional scale in areas of admixture, matching known suture zones in Alpine plants. Simulations using a lattice-based model suggested these empirical patterns arose during or soon after recolonization, long after the estimated origin of the interaction c. 0.5 million years ago. Main conclusions. Species-specific interactions are scarce in alpine habitats because glacial cycles have limited opportunities for coevolution. Their fate, however, remains uncertain under climate change. Here we show that whereas most dispersal routes are paralleled at large scale, regional incongruence implies that the destinies of the species might differ under changing climate. This may be a consequence of the host-dependence of the beetle that locally limits the establishment of dispersing insects.

Landscape structure affects dispersal in the greater white-toothed shrew: inference between genetic and simulated ecological distances

Animal dispersal in a fragmented landscape depends on the complex interaction between landscape structure and animal behavior. To better understand how individuals disperse, it is important to explicitly represent the properties of organisms and the landscape in which they move. A common approach to modelling dispersal includes representing the landscape as a grid of equal sized cells and then simulating individual movement as a correlated random walk. This approach uses a priori scale of resolution, which limits the representation of all landscape features and how different dispersal abilities are modelled. We develop a vector-based landscape model coupled with an object-oriented model for animal dispersal. In this spatially explicit dispersal model, landscape features are defined based on their geographic and thematic properties and dispersal is modelled through consideration of an organism's behavior, movement rules and searching strategies (such as visual cues). We present the model's underlying concepts, its ability to adequately represent landscape features and provide simulation of dispersal according to different dispersal abilities. We demonstrate the potential of the model by simulating two virtual species in a real Swiss landscape. This illustrates the model's ability to simulate complex dispersal processes and provides information about dispersal such as colonization probability and spatial distribution of the organism's path

Effects of cognitive abilities on metapopulation connectivity

Connectivity among demes in a metapopulation depends on both the landscape's and the focal organism's properties (including its mobility and cognitive abilities). Using individual-based simulations, we contrast the consequences of three different cognitive strategies on several measures of metapopulation connectivity. Model animals search suitable habitat patches while dispersing through a model landscape made of cells varying in size, shape, attractiveness and friction. In the blind strategy, the next cell is chosen randomly among the adjacent ones. In the near-sighted strategy, the choice depends on the relative attractiveness of these adjacent cells. In the far-sighted strategy, animals may additionally target suitable patches that appear within their perceptual range. Simulations show that the blind strategy provides the best overall connectivity, and results in balanced dispersal. The near-sighted strategy traps animals into corridors that reduce the number of potential targets, thereby fragmenting metapopulations in several local clusters of demes, and inducing sink-source dynamics. This sort of local trapping is somewhat prevented in the far-sighted strategy. The colonization success of strategies depends highly on initial energy reserves: blind does best when energy is high, near-sighted wins at intermediate levels, and far-sighted outcompetes its rivals at low energy reserves. We also expect strong effects in terms of metapopulation genetics: the blind strategy generates a migrant-pool mode of dispersal that should erase local structures. By contrast, near- and far-sighted strategies generate a propagule-pool mode of dispersal and source-sink behavior that should boost structures (high genetic variance among- and low variance within local clusters of demes), particularly if metapopulation dynamics is also affected by extinction-colonization processes. Our results thus point to important effects of the cognitive ability of dispersers on the connectivity, dynamics and genetics of metapopulations.

Evidence for adaptive radiation from a phylogenetic study of plant defenses.

One signature of adaptive radiation is a high level of trait change early during the diversification process and a plateau toward the end of the radiation. Although the study of the tempo of evolution has historically been the domain of paleontologists, recently developed phylogenetic tools allow for the rigorous examination of trait evolution in a tremendous diversity of organisms. Enemy-driven adaptive radiation was a key prediction of Ehrlich and Raven's coevolutionary hypothesis [Ehrlich PR, Raven PH (1964) Evolution 18:586-608], yet has remained largely untested. Here we examine patterns of trait evolution in 51 North American milkweed species (Asclepias), using maximum likelihood methods. We study 7 traits of the milkweeds, ranging from seed size and foliar physiological traits to defense traits (cardenolides, latex, and trichomes) previously shown to impact herbivores, including the monarch butterfly. We compare the fit of simple random-walk models of trait evolution to models that incorporate stabilizing selection (Ornstein-Ulenbeck process), as well as time-varying rates of trait evolution. Early bursts of trait evolution were implicated for 2 traits, while stabilizing selection was implicated for several others. We further modeled the relationship between trait change and species diversification while allowing rates of trait evolution to vary during the radiation. Species-rich lineages underwent a proportionately greater decline in latex and cardenolides relative to species-poor lineages, and the rate of trait change was most rapid early in the radiation. An interpretation of this result is that reduced investment in defensive traits accelerated diversification, and disproportionately so, early in the adaptive radiation of milkweeds.

Modélisation par automate cellulaire des phénomènes diagénétiques des plateformes carbonatées : calibration et paramétrage à partir de deux cas d'études : l'Urgonien du Vercors (Crétacé inférieur, SE France) et les Calcaires Gris du Mont Compomolon (Lias, NE Italie)

Une fois déposé, un sédiment est affecté au cours de son enfouissement par un ensemble de processus, regroupé sous le terme diagenèse, le transformant parfois légèrement ou bien suffisamment pour le rendre méconnaissable. Ces modifications ont des conséquences sur les propriétés pétrophysiques qui peuvent être positives ou négatives, c'est-à-dire les améliorer ou bien les détériorer. Une voie alternative de représentation numérique des processus, affranchie de l'utilisation des réactions physico-chimiques, a été adoptée et développée en mimant le déplacement du ou des fluides diagénétiques. Cette méthode s'appuie sur le principe d'un automate cellulaire et permet de simplifier les phénomènes sans sacrifier le résultat et permet de représenter les phénomènes diagénétiques à une échelle fine. Les paramètres sont essentiellement numériques ou mathématiques et nécessitent d'être mieux compris et renseignés à partir de données réelles issues d'études d'affleurements et du travail analytique effectué. La représentation des phénomènes de dolomitisation de faible profondeur suivie d'une phase de dédolomitisation a été dans un premier temps effectuée. Le secteur concerne une portion de la série carbonatée de l'Urgonien (Barrémien-Aptien), localisée dans le massif du Vercors en France. Ce travail a été réalisé à l'échelle de la section afin de reproduire les géométries complexes associées aux phénomènes diagénétiques et de respecter les proportions mesurées en dolomite. De plus, la dolomitisation a été simulée selon trois modèles d'écoulement. En effet, la dédolomitisation étant omniprésente, plusieurs hypothèses sur le mécanisme de dolomitisation ont été énoncées et testées. Plusieurs phases de dolomitisation per ascensum ont été également simulées sur des séries du Lias appartenant aux formations du groupe des Calcaire Gris, localisées au nord-est de l'Italie. Ces fluides diagénétiques empruntent le réseau de fracturation comme vecteur et affectent préférentiellement les lithologies les plus micritisées. Cette étude a permis de mettre en évidence la propagation des phénomènes à l'échelle de l'affleurement. - Once deposited, sediment is affected by diagenetic processes during their burial history. These diagenetic processes are able to affect the petrophysical properties of the sedimentary rocks and also improve as such their reservoir capacity. The modelling of diagenetic processes in carbonate reservoirs is still a challenge as far as neither stochastic nor physicochemical simulations can correctly reproduce the complexity of features and the reservoir heterogeneity generated by these processes. An alternative way to reach this objective deals with process-like methods, which simplify the algorithms while preserving all geological concepts in the modelling process. The aim of the methodology is to conceive a consistent and realistic 3D model of diagenetic overprints on initial facies resulting in petrophysical properties at a reservoir scale. The principle of the method used here is related to a lattice gas automata used to mimic diagenetic fluid flows and to reproduce the diagenetic effects through the evolution of mineralogical composition and petrophysical properties. This method developed in a research group is well adapted to handle dolomite reservoirs through the propagation of dolomitising fluids and has been applied on two case studies. The first study concerns a mid-Cretaceous rudist and granular platform of carbonate succession (Urgonian Fm., Les Gorges du Nan, Vercors, SE France), in which several main diagenetic stages have been identified. The modelling in 2D is focused on dolomitisation followed by a dédolomitisation stage. For the second study, data collected from outcrops on the Venetian platform (Lias, Mont Compomolon NE Italy), in which several diagenetic stages have been identified. The main one is related to per ascensum dolomitisation along fractures. In both examples, the evolution of the effects of the mimetic diagenetic fluid on mineralogical composition can be followed through space and numerical time and help to understand the heterogeneity in reservoir properties. Carbonates, dolomitisation, dédolomitisation, process-like modelling, lattice gas automata, random walk, memory effect.

A high-mobility, low-cost phenotype defines human effector-memory CD8+ T cells.

T cells move randomly ("random-walk"), a characteristic thought to be integral to their function. Using migration assays and time-lapse microscopy, we found that CD8+ T cells lacking the lymph node homing receptors CCR7 and CD62L migrate more efficiently in transwell assays, and that these same cells are characterized by a high frequency of cells exhibiting random crawling activity under culture conditions mimicking the interstitial/extravascular milieu, but not when examined on endothelial cells. To assess the energy efficiency of cells crawling at a high frequency, we measured mRNA expression of genes key to mitochondrial energy metabolism (peroxisome proliferator-activated receptor gamma coactivator 1beta [PGC-1beta], estrogen-related receptor alpha [ERRalpha], cytochrome C, ATP synthase, and the uncoupling proteins [UCPs] UCP-2 and -3), quantified ATP contents, and performed calorimetric analyses. Together these assays indicated a high energy efficiency of the high crawling frequency CD8+ T-cell population, and identified differentially regulated heat production among nonlymphoid versus lymphoid homing CD8+ T cells.

Linear random knots and their scaling behavior

We present here a nonbiased probabilistic method that allows us to consistently analyze knottedness of linear random walks with up to several hundred noncorrelated steps. The method consists of analyzing the spectrum of knots formed by multiple closures of the same open walk through random points on a sphere enclosing the walk. Knottedness of individual "frozen" configurations of linear chains is therefore defined by a characteristic spectrum of realizable knots. We show that in the great majority of cases this method clearly defines the dominant knot type of a walk, i.e., the strongest component of the spectrum. In such cases, direct end-to-end closure creates a knot that usually coincides with the knot type that dominates the random closure spectrum. Interestingly, in a very small proportion of linear random walks, the knot type is not clearly defined. Such walks can be considered as residing in a border zone of the configuration space of two or more knot types. We also characterize the scaling behavior of linear random knots.

Plasma cell precursors: long-distance travelers looking for a home.

Little is known about the migration of plasma cell precursors to the lymph node medulla. In this issue of Immunity, Fooksman et al. (2010) propose that this migration is largely independent of chemotactic cues but follows a long linear walk of random orientation.