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In this paper, we study the dynamic risk measures for processes induced by backward stochastic differential equations driven by Teugel's martingales associated with Lévy processes (BSDELs). The representation theorem for generators of BSDELs is provided. Furthermore, the time consistency of the coherent and convex dynamic risk measures for processes is characterized by means of the generators of BSDELs. Moreover, the coherency and convexity of dynamic risk measures for processes are characterized by the generators of BSDELs. Finally, we provide two numerical examples to illustrate the proposed dynamic risk measures.

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Fokker-Planck PDEs (including diffusions) for stable Lévy processes (including Wiener processes) on the joint space of positions and orientations play a major role in mechanics, robotics, image analysis, directional statistics and probability theory. Exact analytic designs and solutions are known in the 2D case, where they have been obtained using Fourier transform on S E ( 2 ) . Here, we extend these approaches to 3D using Fourier transform on the Lie group S E ( 3 ) of rigid body motions. More precisely, we define the homogeneous space of 3D positions and orientations R 3 ⋊ S 2 : = S E ( 3 ) / ( { 0 } × S O ( 2 ) ) as the quotient in S E ( 3 ) . In our construction, two group elements are equivalent if they are equal up to a rotation around the reference axis. On this quotient, we design a specific Fourier transform. We apply this Fourier transform to derive new exact solutions to Fokker-Planck PDEs of α -stable Lévy processes on R 3 ⋊ S 2 . This reduces classical analysis computations and provides an explicit algebraic spectral decomposition of the solutions. We compare the exact probability kernel for α = 1 (the diffusion kernel) to the kernel for α = 1 2 (the Poisson kernel). We set up stochastic differential equations (SDEs) for the Lévy processes on the quotient and derive corresponding Monte-Carlo methods. We verified that the exact probability kernels arise as the limit of the Monte-Carlo approximations.

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Recently, the scaling limit of cluster sizes for critical inhomogeneous random graphs of rank-1 type having finite variance but infinite third moment degrees was obtained in Bhamidi et al. (Ann Probab 40:2299-2361, 2012). It was proved that when the degrees obey a power law with exponent τ ∈ ( 3 , 4 ) , the sequence of clusters ordered in decreasing size and multiplied through by n - ( τ - 2 ) / ( τ - 1 ) converges as n → ∞ to a sequence of decreasing non-degenerate random variables. Here, we study the tails of the limit of the rescaled largest cluster, i.e., the probability that the scaling limit of the largest cluster takes a large value u, as a function of u. This extends a related result of Pittel (J Combin Theory Ser B 82(2):237-269, 2001) for the Erdos-Rényi random graph to the setting of rank-1 inhomogeneous random graphs with infinite third moment degrees. We make use of delicate large deviations and weak convergence arguments.

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The extent to which a cognitive system's behavioral dynamics fit a power law distribution is considered indicative of the extent to which that system's behavior is driven by multiplicative, interdependent interactions between its components. Here, we investigate the dynamics of memory processes in individual and collaborating participants. Collaborative dyads showed the characteristic collaborative inhibition effect when compared to nominal groups in terms of the number of items retrieved in a categorical recall task, but they also generate qualitatively different patterns of search behavior. To categorize search behavior, we used multi-model inference to compare the degree to which five candidate models (normal, exponential, gamma, lognormal, and Pareto) described the temporal distribution of each individual and dyad's recall processes. All individual and dyad recall processes were best fit by interaction-dominant distributions (lognormal and Pareto), but a clear difference emerged in that individual behavior is more power law, and collaborative behavior was more lognormal. We discuss these results in terms of the cocktail model (Holden et al. in Psychol Rev 116(2):318-342, 2009), which suggests that as a task becomes more constrained (such as through the necessity of collaborating), behavior can shift from power law to lognormal. This shift may reflect a decrease in the dyad's ability to flexibly shift between perseverative and explorative search patterns. Finally, our results suggest that a fruitful avenue for future research would be to investigate the constraints modulating the shift from power law to lognormal behavior in collaborative memory search.

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Several studies have reported that fragmentation (e.g. of anthropogenic origin) of habitats often leads to a decrease in the number of species in the region. An important mechanism causing this adverse ecological impact is the change in the encounter rates (i.e. the rates at which individuals meet other organisms of the same or different species). Yet, how fragmentation can change encounter rates is poorly understood. To gain insight into the problem, here we ask how landscape fragmentation affects encounter rates when all other relevant variables remain fixed. We present strong numerical evidence that fragmentation decreases search efficiencies thus encounter rates. What is surprising is that it falls even when the global average densities of interacting organisms are held constant. In other words, fragmentation per se can reduce encounter rates. As encounter rates are fundamental for biological interactions, it can explain part of the observed diminishing in animal biodiversity. Neglecting this effect may underestimate the negative outcomes of fragmentation. Partial deforestation and roads that cut through forests, for instance, might be responsible for far greater damage than thought. Preservation policies should take into account this previously overlooked scientific fact.

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