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A principal problem of evolutionary graph theory is to find the probability that an initial mutant population will fix on a graph, i.e. that the mutants will eventually replace the indigenous population. This problem is particularly difficult when the dimensionality of a graph is high. Martingales can yield compact and exact expressions for the fixation probability of an evolutionary graph. Crucially, the tractability of martingales does not necessarily depend on the dimensionality of a graph. We will use martingales to obtain the exact fixation probability of graphs with high dimensionality, specifically k-partite graphs (or 'circular flows') and megastars (or 'superstars'). To do so, we require that the edges of the graph permit mutants to reproduce in one direction and indigenous in the other. The resultant expressions for fixation probabilities explicitly show their dependence on the parameters that describe the graph structure, and on the starting position(s) of the initial mutant population. In particular, we will investigate the effect of funneling on the fixation probability of k-partite graphs, as well as the effect of placing an initial mutant in different partitions. These are the first exact and explicit results reported for the fixation probability of evolutionary graphs with dimensionality greater than 2, that are valid over all parameter space. It might be possible to extend these results to obtain fixation probabilities of high-dimensional evolutionary graphs with undirected or directed connections. Martingales are a formidable theoretical tool that can solve fundamental problems in evolutionary graph theory, often within a few lines of straightforward mathematics., (Copyright © 2018 Elsevier Ltd. All rights reserved.)

Biological memory is a ubiquitous function that can generate a sustained response to a transient inductive stimulus. To better understand this function, we must consider the mechanisms by which different structures of genetic networks achieve memory. Here, we investigated two competitive gene regulatory network models: the regulated mutual activation network (MAN) and the regulated mutual repression network (MRN). Stochasticity deteriorated the persistence of memory of both the MAN and the MRN. Mathematical comparison by simulation and theoretical analysis identified functional differences in the stochastic memory between the competitive models: specifically, the MAN provided much more robust, persistent memory than the MRN. The stochastic persistent memory pattern of the MAN can be adjusted by changing the binding strength of the activators, whereas the MRN required highly cooperative and strong binding repressors for robust memory., (Copyright © 2017 Elsevier Ltd. All rights reserved.)

It has been proposed that the development of verbal counting is supported by a more ancient preverbal system of estimation, the most widely canvassed candidates being the accumulator originally proposed by Gibbon and colleagues and the analogue magnitude system proposed by Dehaene and colleagues. The aim of this chapter is to assess the strengths and weaknesses of these models in terms of their capacity to emulate the statistical properties of verbal counting. The emphasis is put on the emergence of exact representations, autoscaling, and commensurability of noise characteristics. We also outline the modified architectures that may help improve models' power to meet these criteria. We propose that architectures considered in this chapter can be used to generate predictions for experimental testing and provide an example where we test the hypothesis whether the visual sense of number, ie, ability to discriminate numerosity without counting, entails enumeration of objects., (© 2016 Elsevier B.V. All rights reserved.)

Loss is an important parameter of Quality of Service (QoS). Though stochastic network calculus is a very useful tool for performance evaluation of computer networks, existing studies on stochastic service guarantees mainly focused on the delay and backlog. Some efforts have been made to analyse loss by deterministic network calculus, but there are few results to extend stochastic network calculus for loss analysis. In this paper, we introduce a new parameter named loss factor into stochastic network calculus and then derive the loss bound through the existing arrival curve and service curve via this parameter. We then prove that our result is suitable for the networks with multiple input flows. Simulations show the impact of buffer size, arrival traffic, and service on the loss factor.

Tychastic viability is defined in an uncertain dynamical framework and used for providing a "viability risk eradication measure", first, by delineating the set of initial conditions from which all evolutions satisfy viability constraints, second, for the other "risky" initial states, by introducing their duration index. This approach provides an alternative to the stochastic representation of chance and these two measures replace the statistical measures (expectation, variance, etc).

In a random number generation task, participants are asked to generate a random sequence of numbers, most typically the digits 1 to 9. Such number sequences are not mathematically random, and both extent and type of bias allow one to characterize the brain's "internal random number generator". We assume that certain patterns and their variations will frequently occur in humanly generated random number sequences. Thus, we introduce a pattern-based analysis of random number sequences. Twenty healthy subjects randomly generated two sequences of 300 numbers each. Sequences were analysed to identify the patterns of numbers predominantly used by the subjects and to calculate the frequency of a specific pattern and its variations within the number sequence. This pattern analysis is based on the Damerau-Levenshtein distance, which counts the number of edit operations that are needed to convert one string into another. We built a model that predicts not only the next item in a humanly generated random number sequence based on the item's immediate history, but also the deployment of patterns in another sequence generated by the same subject. When a history of seven items was computed, the mean correct prediction rate rose up to 27% (with an individual maximum of 46%, chance performance of 11%). Furthermore, we assumed that when predicting one subject's sequence, predictions based on statistical information from the same subject should yield a higher success rate than predictions based on statistical information from a different subject. When provided with two sequences from the same subject and one from a different subject, an algorithm identifies the foreign sequence in up to 88% of the cases. In conclusion, the pattern-based analysis using the Levenshtein-Damarau distance is both able to predict humanly generated random number sequences and to identify person-specific information within a humanly generated random number sequence.

Lognormal statistical distributions are observed in a variety of scientific fields. The widths of these distributions in the log scale are often similar, but the underlying mechanism that maintains these widths within a small range has not been well explained. We show that a stochastic process of halving followed by addition can yield a stationary distribution that resembles the universal lognormal distribution with a certain width. The mechanism that we propose here would provide insight into the essence of why lognormal-like distributions in many systems have a common width.

The birth/death process with mutation describes the evolution of a population, and displays rich dynamics including clustering and fluctuations. We discuss an analytical 'field-theoretical' approach to the birth/death process, using a simple dimensional analysis argument to describe evolution as a 'super-Brownian motion' in the infinite population limit. The field theory technique provides corrections to this for large but finite population, and an exact description at arbitrary population size. This allows a characterisation of the difference between the evolution of a phenotype, for which strong local clustering is observed, and a genotype for which distributions are more dispersed. We describe the approach with sufficient detail for non-specialists.

We compare and contrast the long-time dynamical properties of two individual-based models of biological coevolution. Selection occurs via multispecies, stochastic population dynamics with reproduction probabilities that depend nonlinearly on the population densities of all species resident in the community. New species are introduced through mutation. Both models are amenable to exact linear stability analysis, and we compare the analytic results with large-scale kinetic Monte Carlo simulations, obtaining the population size as a function of an average interspecies interaction strength. Over time, the models self-optimize through mutation and selection to approximately maximize a community potential function, subject only to constraints internal to the particular model. If the interspecies interactions are randomly distributed on an interval including positive values, the system evolves toward self-sustaining, mutualistic communities. In contrast, for the predator-prey case the matrix of interactions is antisymmetric, and a nonzero population size must be sustained by an external resource. Time series of the diversity and population size for both models show approximate 1/f noise and power-law distributions for the lifetimes of communities and species. For the mutualistic model, these two lifetime distributions have the same exponent, while their exponents are different for the predator-prey model. The difference is probably due to greater resilience toward mass extinctions in the food-web like communities produced by the predator-prey model.

At present two growth models describe successfully the distribution of size and topological complexity in populations of dendritic trees with considerable accuracy and simplicity, the BE model (Van Pelt et al. in J. Comp. Neurol. 387:325-340, 1997) and the S model (Van Pelt and Verwer in Bull. Math. Biol. 48:197-211, 1986). This paper discusses the mathematical basis of these models and analyzes quantitatively the relationship between the BE model and the S model assumed in the literature by developing a new explicit equation describing the BES model (a dendritic growth model integrating the features of both preceding models; Van Pelt et al. in J. Comp. Neurol. 387:325-340, 1997). In numerous studies it is implicitly presupposed that the S model is conditionally linked to the BE model (Granato and Van Pelt in Brain Res. Dev. Brain Res. 142:223-227, 2003; Uylings and Van Pelt in Network 13:397-414, 2002; Van Pelt, Dityatev and Uylings in J. Comp. Neurol. 387:325-340, 1997; Van Pelt and Schierwagen in Math. Biosci. 188:147-155, 2004; Van Pelt and Uylings in Network. 13:261-281, 2002; Van Pelt, Van Ooyen and Uylings in Modeling Dendritic Geometry and the Development of Nerve Connections, pp 179, 2000). In this paper we prove the non-exactness of this assumption, quantify involved errors and determine the conditions under which the BE and S models can be separately used instead of the BES model, which is more exact but considerably more difficult to apply. This study leads to a novel expression describing the BE model in an analytical closed form, much more efficient than the traditional iterative equation (Van Pelt et al. in J. Comp. Neurol. 387:325-340, 1997) in many neuronal classes. Finally we propose a new algorithm in order to obtain the values of the parameters of the BE model when this growth model is matched to experimental data, and discuss its advantages and improvements over the more commonly used procedures.

Given a set of third- or higher-order moments, not only is the saddlepoint approximation the only realistic 'family-free' technique available for constructing an associated probability distribution, but it is 'optimal' in the sense that it is based on the highly efficient numerical method of steepest descents. However, it suffers from the problem of not always yielding full support, and whilst [S. Wang, General saddlepoint approximations in the bootstrap, Prob. Stat. Lett. 27 (1992) 61.] neat scaling approach provides a solution to this hurdle, it leads to potentially inaccurate and aberrant results. We therefore propose several new ways of surmounting such difficulties, including: extending the inversion of the cumulant generating function to second-order; selecting an appropriate probability structure for higher-order cumulants (the standard moment closure procedure takes them to be zero); and, making subtle changes to the target cumulants and then optimising via the simplex algorithm.

A nonlinear stochastic model for the dynamics of a population with either a continuous size structure or a discontinuous stage structure is formulated in the Eulerian formalism. It takes into account dispersion effects due to stochastic variability of the development process of the individuals. The discrete equations of the numerical approximation are derived, and an analysis of the existence and stability of the equilibrium states is performed. An application to a copepod population is illustrated; numerical results of Eulerian and Lagrangian models are compared.

We study the (Ito) stochastic differential equation [equation: see text] as a model for population growth in a stochastic environment with finite carrying capacity K > 0. Here r and alpha are constants and Bt denotes Brownian motion. If r > or = 0, we show that this equation has a unique strong global solution for all x > 0 and we study some of its properties. Then we consider the following problem: What harvesting strategy maximizes the expected total discounted amount harvested (integrated over all future times)? We formulate this as a stochastic control problem. Then we show that there exists a constant optimal "harvest trigger value" x* epsilon (0, K) such that the optimal strategy is to do nothing if Xt < x* and to harvest Xt-x* if Xt > x*. This leads to an optimal population process Xt being reflected downward at x*. We find x* explicitly.

The aim of this study is to derive an asymptotic expression for the probability that an infectious disease will disappear from a population at the end of a major outbreak ('fade-out'). The study deals with a stochastic SIR-model. Local asymptotic expansions are constructed for the deterministic trajectories of the corresponding deterministic system, in particular for the deterministic trajectory starting in the saddle point. The analytical expression for the probability of extinction is derived by asymptotically solving a boundary value problem based on the Fokker-Planck equation for the stochastic system. The asymptotic results are compared with results obtained by random walk simulations.

It has been shown that the density function of the duration of infection in a superinfection malaria model can be approximated by a mixture of exponential density functions. This is achieved by an application of continued fractions. Numerical calculations and graphs illustrate that this approach is effective.

A new model of molecular evolution is introduced that allows for heterogeneous rates across the sequence positions. The development of this model was motivated by two issues: first, a number of studies have shown that the positions in a DNA sequence evolve at different rates, and second, it has been shown that not accounting for this heterogeneity can lead to biased estimates of evolutionary parameters. The authors generalize the Markovian model of molecular evolution to allow for heterogeneous rates and explore some of the consequences of such a model. In particular, they quantify the biases incurred by incorrectly assuming an equal-rate model and consider what can be learned about evolutionary parameters under a heterogeneous model.

A new stochastic model for the residence time distribution of a drug injected instantaneously into the circulatory system is proposed and analyzed. The properties of the residence time are derived from the assumptions made about the cycle time distribution and the rule for elimination. This rule is given by the probability distribution of the number of cycles needed for elimination of a randomly selected molecule of the drug. Only the geometric distribution has been previously used for this purpose. Its transformation is applied here to get a boundary for the residence time. Other discrete distributions are applied with a view to describing different experimental situations. Suitable continuous probability distributions for the cycle time description are discussed.

This paper deals with a new class of stochastic species abundance models where the abundances are the points of an inhomogeneous Poisson process. These models are the result of a dynamic approach in which the changes in abundances through time are described by a multivariate diffusion and speciation constitutes a homogeneous Poisson process. In particular, the lognormal model is generated by assuming that the density regulation within each species is given by the Gompertz curve and that the environmental variances are constant. A substantial generalization is obtained by introducing a general type of interspecific density regulation and correlated environmental noise. This more general mechanism also generates the lognormal species abundance distribution.

In this paper we develop a dynamical theory of coevolution in ecological communities. The derivation explicitly accounts for the stochastic components of evolutionary change and is based on ecological processes at the level of the individual. We show that the coevolutionary dynamic can be envisaged as a directed random walk in the community's trait space. A quantitative description of this stochastic process in terms of a master equation is derived. By determining the first jump moment of this process we abstract the dynamic of the mean evolutionary path. To first order the resulting equation coincides with a dynamic that has frequently been assumed in evolutionary game theory. Apart from recovering this canonical equation we systematically establish the underlying assumptions. We provide higher order corrections and show that these can give rise to new, unexpected evolutionary effects including shifting evolutionary isoclines and evolutionary slowing down of mean paths as they approach evolutionary equilibria. Extensions of the derivation to more general ecological settings are discussed. In particular we allow for multi-trait coevolution and analyze coevolution under nonequilibrium population dynamics.

The proposed mathematical model investigates the simplified cytomechanics of cell shape change driven by stochastic stimulation from chemosensory receptors. The cytomechanical component of our model describes the dynamical distribution of F-actin and associated forces in an idealized cortical actin network around the cell periphery. The chemosensory component describes the distribution of chemotactic receptors in the cell membrane surrounding the cortex, where bound receptors give rise to an intracellular signal which modulates some property of the cortical network. As in our earlier models, an account is made for (1) the reactive, contractive properties of cortical actin, but here also for a stress induced by curvature of the cortex-membrane complex which carries an effective surface tension, and (2) statistical fluctuations in receptor binding, but generalized here to include statistical fluctuations in the spatial distribution of receptors, entirely determined by the additional prescription of membrane diffusion coefficients along with total receptor number, receptor binding rate constants and the local concentration field of chemotactic factor. We simplify the analysis by restricting the model to a prototype in which viscous stresses in the cortical network are negligible and the radial extension of the cell cortex is a prescribed function of the cortical actin concentration. We assume in particular that the assembly rate of cortical actin depends on the local density of bound receptors. These assumptions lead to a 4th-order parabolic differential equation on the unit circle coupled to a system of stochastic differential equations. We characterize via bifurcation analysis, stochastic simulations, and analytical correlation functions the spatial-temporal pattern of cell morphology under the influence of fluctuations in the bound receptor distribution for the case of a uniform concentration field of chemotactic factor. In addition to addressing the biological significance of our model, we remark on its relevance to the generic problem of the influence of correlated stochastic perturbations on spatial patterns in morphogenetic media.

Incoherent quasi-elastic neutron scattering is applied to study the local diffusion and chain dynamics of L-alpha-diplamiotylphosphatidylcholine molecules in oriented model membranes. Different motions are distinguished by changing the hydration of the multilayers as well as by measuring below and above the gel-to-liquid crystalline phase transition. The time range of the utilized time-of-flight spectrometer permits to observe two types of motion to be observed more closely: chain defect motions and the local diffusion of the whole molecule in its solvation cage. Oriented lipid membranes are a useful system for the observation of chain defects, as they can be macroscopically oriented, in contrast to most polymers. As a representative model for a chain defect a kink is chosen and the corresponding scattering functions are derived. The kink motion can explain the entire dynamics seen in the gel phase, and the lifetime of such a defect was found to be 10-15 ps, in good agreement with theoretical predictions. On the other hand the dynamics in the liquid crystalline phase cannot be explained even by a superposition of several kinks and thus requires the consideration of an additional motion: the local diffusion of the molecule in its solvation cage. The size of the solvation cage is increasing with multilayer hydration and reduced temperature. Particularly interesting in view of recent discussions about the origin of the short-range repulsive forces between membranes is the experimental finding of an out-of-plane motion with an amplitude of 1-1.5 A, which cannot be explained by the undulation of the whole membrane.

We consider a stochastic mechanism of the loss of resistance of cancer cells to cytotoxic agents, in terms of unstable gene amplification. Two models being different versions of a time-continuous branching random walk are presented. Both models assume strong dependence in replication and segregation of the extrachromosomal elements. The mathematical part of the paper includes the expression for the expected number of cells with a given number of gene copies in terms of modified Bessel functions. This adds to the collection of rare explicit solutions to branching process models. Original asymptotic expansions are also demonstrated. Fitting the model to experimental data yields estimates of the probabilities of gene amplification and deamplification. The thesis of the paper is that purely stochastic mechanisms may explain the dynamics of reversible drug resistance of cancer cells. Various stochastic approaches and their limitations are discussed.

Cross diffusion has been widely considered in the mathematical modelling of spatially structured ecological and epidemic systems, either in the mechanical description of diffusion or in the stochastic point process description of interacting populations. In this paper mathematical results recently obtained by the authors about the asymptotic behaviour of reaction-diffusion systems with full matrices of diffusion are applied to classes of biological systems.

In summary, Table II is a listing of the pitfalls and advantages of using Monte Carlo simulations and numerical solution of the diffusion equation to describe photon migration in tissues. Judicious use of these techniques to describe the solution to the forward imaging problem may allow determination of the best theoretical resolution and the smallest detectable volume for the range of optical property differences expected in situ or imposed by contrast agent administration. Furthermore, an understanding of the forward imaging problem through these numerical techniques also contributes to our understanding of the most efficient solution to the inverse imaging problem.

Mathematical techniques are presented for the transformation of digital anatomical textbooks from the ideal to the individual, allowing for the representation of the variabilities manifest in normal human anatomies. The ideal textbook is constructed on a fixed coordinate system to contain all of the information currently available about the physical properties of neuroanatomies. This information is obtained via sensor probes such as magnetic resonance, as well as computed axial and emission tomography, along with symbolic information such as white- and gray-matter tracts, nuclei, etc. Human variability associated with individuals is accommodated by defining probabilistic transformations on the textbook coordinate system, the transformations forming mathematical translation groups of high dimension. The ideal is applied to the individual patient by finding the transformation which is consistent with physical properties of deformable elastic solids and which brings the coordinate system of the textbook to that of the patient. Registration, segmentation, and fusion all result automatically because the textbook carries symbolic values as well as multisensor features.

The active migration of blood and tissue cells is important in a number of physiological processes including inflammation, wound healing, embryogenesis, and tumor cell metastasis. These cells move by transmitting cytoplasmic force through membrane receptors which are bound specifically to adhesion ligands in the surrounding substratum. Recently, much research has focused on the influence of the composition of extracellular matrix and the distribution of its components on the speed and direction of cell migration. It is commonly believed that the magnitude of the adhesion influences cell speed and/or random turning behavior, whereas a gradient of adhesion may bias the net direction of the cell movement, a phenomenon known as haptotaxis. The mechanisms underlying these responses are presently not understood. A stochastic model is presented to provide a mechanistic understanding of how the magnitude and distribution of adhesion ligands in the substratum influence cell movement. The receptor-mediated cell migration is modeled as an interrelation of random processes on distinct time scales. Adhesion receptors undergo rapid binding and transport, resulting in a stochastic spatial distribution of bound receptors fluctuating about some mean distribution. This results in a fluctuating spatio-temporal pattern of forces on the cell, which in turn affects the speed and turning behavior on a longer time scale. The model equations are a system of nonlinear stochastic differential equations (SDE's) which govern the time evolution of the spatial distribution of bound and free receptors, and the orientation and position of the cell. These SDE's are integrated numerically to simulate the behavior of the model cell on both a uniform substratum, and on a gradient of adhesion ligand concentration. Furthermore, analysis of the governing SDE system and corresponding Fokker-Planck equation (FPE) yields analytical expressions for indices which characterize cell movement on multiple time scales in terms of cell cytomechanical, morphological, and receptor binding and transport parameters. For a uniform adhesion ligand concentration, this analysis provides expressions for traditional cell movement indices such as mean speed, directional persistence time, and random motility coefficient. In a small gradient of adhesion, a perturbation analysis of the FPE yields a constitutive cell flux expression which includes a drift term for haptotactic directional cell migration. The haptotactic drift contains terms identified as contributions from directional orientation bias (taxis), kinesis, and orthotaxis, of which taxis appears to be predominant given estimates of the model parameters.

While the mathematical theory of epidemics has its origins with Ross (1911), it was not until Kryscio (1975) that explicit expressions for the state probabilities of the classical general epidemic model established by Bartlett (1949) were found. However, these formulae were of limited practical use when the population size was of even moderate size. By shifting the focus from the bivariate pair representing the number of susceptibles and infectives to that for the number of infectives and removals, one is able to obtain solutions that are considerably simpler and easier to manage than those previously derived and which are not restricted by the size of the population. The results are obtained for a generalized general epidemic process in which transition probabilities are arbitrary functions of the state space, and then applied to the classical model. An extension to time-dependent transition rates is also considered.

Three stochastic eye-movement models for arithmetic and reading performance have been proposed, one for arithmetic and two for reading. Each model characterizes a real-time stochastic process in terms of fixation durations and saccadic movement, but only direction and length of saccades are considered, not acceleration or velocity. Aspects of the models that are emphasized, partly because of their general neglect in the literature, are the probability distribution of fixation durations and the random walk of saccade directions. The distributions of fixation duration are approximately exponential, but systematic deviations can be accounted for in the models, even though the fit to data is not perfect. In the case of the arithmetic algorithms of addition and subtraction, the random walk of the normative model has only two possible moves. Data are also presented on backtracking, skipping and wandering eye movements, each of which has a significant relative frequency. The first reading model is called a minimal control model, because it does not take account of the effects of many local variables, e.g., word length, that have been extensively studied. The axioms on fixation duration for the minimal control model are the same as for the arithmetic model. Abstracting from the different arrangement of stimuli in arithmetic algorithms and in linear text, the axioms on saccadic motion for the two models are also essentially identical. The stochastic nature of both models is strongly supported by data on the independence of fixation durations from previous fixation durations. Additional detailed evidence is presented for the arithmetic model. To better account for a great variety of experimental results concerning significant effects on eye movements in reading, a text-dependent probabilistic model of reading is introduced. Significant local effects fall into three classes, identified as line, word and grammatical variables. The revised axioms embody five features of text known to be significant: (i) fixation duration depends on the number of letters in a word; (ii) a saccade is longer when a longer word is to the right; (iii) a saccade is longer when the current fixation is on a longer word; (iv) high-frequency fixation words have the highest probability of being skipped; (v) ambiguous or difficult grammatical structures increase backtracking.

The random walk function is a mathematical function derived from studies of the mass transport and flow of diffusible materials through tubes. Approximations to the function were first used some time ago in the field of cardiac tracer dilution curves, but in the absence of rapid and reproducible curve fitting the method never became commonplace. The current study uses the latest curve-fitting techniques and shows how the method may be used with precision in the analysis of time-activity curves from dynamic oesophageal and blood flow studies. The physiological basis of the method is given and parameters obtained which relate to both the rate of flow and the local dispersion of the bolus.

In [Trans. Am. Math. Soc. 375 (2022), no. 1, 627–668], Kahn gave the strongest possible, affirmative, answer to Shamir's problem, which had been open since the late 1970s: Let r⩾3$$ r\geqslant 3 $$ and let n$$ n $$ be divisible by r$$ r $$. Then, in the random r$$ r $$‐uniform hypergraph process on n$$ n $$ vertices, as soon as the last isolated vertex disappears, a perfect matching emerges. In the present work, we prove the analogue of this result for clique factors in the random graph process: at the time that the last vertex joins a copy of the complete graph Kr$$ {K}_r $$, the random graph process contains a Kr$$ {K}_r $$‐factor. Our proof draws on a novel sequence of couplings which embeds the random hypergraph process into the cliques of the random graph process. An analogous result is proved for clique factors in the s$$ s $$‐uniform hypergraph process (s⩾3$$ s\geqslant 3 $$). [ABSTRACT FROM AUTHOR]

We present new high order approximations schemes for the Cox–Ingersoll–Ross (CIR) process that are obtained by using a recent technique developed by Alfonsi and Bally (2021, A generic construction for high order approximation schemes of semigroups using random grids. Numer. Math. , 148 , 743–793) for the approximation of semigroups. The idea consists in using a suitable combination of discretization schemes calculated on different random grids to increase the order of convergence. This technique coupled with the second order scheme proposed by Alfonsi (2010, High order discretization schemes for the CIR process: application to affine term structure and Heston models. Math. Comp. , 79 , 209–237) for the CIR leads to weak approximations of order |$2k$|⁠ , for all |$k\in{{\mathbb{N}}}^{\ast }$|⁠. Despite the singularity of the square-root volatility coefficient, we show rigorously this order of convergence under some restrictions on the volatility parameters. We illustrate numerically the convergence of these approximations for the CIR process and for the Heston stochastic volatility model and show the computational time gain they give. [ABSTRACT FROM AUTHOR]

Inspired by the work by Brychkov and Saad who gave infinite summation formulas for Appell functions (of two variables) with the help of summation theorems, we, using a new inverse pair of symbolic operators, establish infinite summation formulas for four Lauricella functions of three variables. As an application of the results obtained, we present integral representation formulas. Bibliography:30titles., 1 Introduction and Definitions Hypergeometric functions of one and more variables naturally occur in a wide variety of problems in applied mathematics, statistics, operations research, theoretical physics, and engineering sciences. [...]

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Unless an appropriate dissipation mechanism is introduced in its evolution, a deterministic system generally does not tend to equilibrium. However, coarse-graining such a system implies a mesoscopic representation which is no longer deterministic. The mesoscopic system should be addressed by stochastic methods, but they lead to practically infeasible calculations. However, following the pioneering work of Kolmogorov, one finds that such mesoscopic systems can be approximated by Markov processes in relevant conditions, mainly, if the microscopic system is ergodic. So, the mesoscopic system tends to stationarity in specific situations, as expected from thermodynamics. Kolmogorov proved that in the stationary case, the instantaneous entropy of the mesoscopic process, conditioned by its past trajectory, tends to a finite limit at infinite times. Thus, one can define the Kolmogorov entropy. It can be shown that in certain situations, this property remains true even in the nonstationary case. We anticipated this important conclusion in a previous article, giving some elements of a justification, whereas it is precisely derived below in relevant conditions and in the case of a discrete system. It demonstrates that the Kolmogorov entropy is linked to basic aspects of time, such as its irreversibility. This extends the well-known conclusions of Boltzmann and of more recent researchers and gives a general insight to the fascinating relation between time and entropy.

Abstract Hamilton–Jacobi–Bellman equation (HJBE) and backward stochastic differential equation (BSDE) are the two faces of stochastic control. We explore their equivalence focusing on a system of self-exciting and affine stochastic differential equations (SDEs) arising in streamflow dynamics. Our SDE is a finite-dimensional Markovian embedding of an infinite-dimensional jump-driven process called the superposition of continuous-state branching processes (a supCBI process). We formulate new ergodic control problems to evaluate the worst-case streamflow discharge in the long run and derive their HJBEs and ergodic BSDEs. The constant ambiguity aversion classically used in assessing model ambiguity must be modified in our case so that the optimality equations become well-posed. With a suitable modification of the ambiguity-aversion coefficient depending on the distributed reversion speed, we demonstrate that the solutions to the optimality equations are equivalent to each other in the sense that they lead to the same result. Finally, we apply the proposed framework to the computation of realistic cases with an existing record of discharge through a numerical Markovian embedding.

A very important class of models widely used nowadays to describe and predict, at least in stochastic terms, the behavior of many-particle systems (where the word “particle” is not meant in the purely mechanical sense: particles can be cells of a living tissue, or cars in a traffic flow, or even members of an animal or human population) is the Kinetic Theory for Active Particles, i.e., a scheme of possible generalizations and re-interpretations of the Boltzmann equation. Now, though in the literature on the subject this point is systematically disregarded, this scheme is based on Markov Chains, which are special stochastic processes with important properties they share with many natural processes. This circumstance is here carefully discussed not only to suggest the different ways in which Markov Chains can intervene in equations describing the stochastic behavior of any many-particle system, but also, as a preliminary methodological step, to point out the way in which the notion of a Markov Chain can be suitably generalized to this aim. As a final result of the discussion, we find how to develop new very plausible and likely ways to take into account possible effects of the external world on a non-isolated many-particle system, with particular attention paid to socio-economic problems.

This study presents a novel event-triggered relearning framework for neural network modeling, designed to improve prediction precision in dynamic stochastic complex industrial systems under non-stationary and variable conditions. Firstly, a sliding window algorithm combined with entropy is applied to divide the input and output datasets across different operational conditions, establishing clear data boundaries. Following this, the prediction errors derived from the neural network under different operational states are harnessed to define a set of event-triggered relearning criteria. Once these conditions are triggered, the relevant dataset is used to recalibrate the model to the specific operational condition and predict the data under this operating condition. When the predicted data fall within the training input range of a pre-trained model, we switch to that model for immediate prediction. Compared with the conventional BP neural network model and random vector functional-link network, the proposed model can produce a better estimation accuracy and reduce computation costs. Finally, the effectiveness of our proposed method is validated through numerical simulation tests using nonlinear Hammerstein models with Gaussian noise, reflecting complex stochastic industrial processes.

Let [Please download the PDF to view the mathematical expression] be a weighted sum with independent, identically distributed steps [X.sub.j,n], j < n, where [a.sub.j,n] = f(j/n) for some f [member of] [C.sup.2] [0,1]. Under Cramer's condition, we prove an integro-local limit theorem for [[Please download the PDF to view the mathematical expression] as x/n [member of] [[m.sup.-],[m.sup.+]] for some [m.sup.-], [m.sup.+] and any sequence [DELTA].sub.n] tending to zero slowly enough. This result covers the whole scope of normal, moderate, and large deviations. For the stochastic process [Y.sub.n](t) corresponding to [S.sub.0],..., [S.sub.n] we obtain a conditional functional limit theorem concerning convergence [Y.sub.n](t) to the Brownian bridge given the condition [S.sub.n] [member of] [x,x + [DELTA]n)., 1. Introduction Let [Y.sub.1,n],..., [Y.sub.n,n] be a triangular array of independent random variables with nonlattice distribution. Normal, moderate, and large deviations of [Please download the PDF to view the mathematical [...]

This paper provides a refinement of the study of asymptotic behaviour for a class of nonlocal in time telegraph equations with positively singular kernels. Based on fundamental properties of relaxation functions and recent representation of the fundamental solution in [Nonlinear Anal. 193 (2020), 111411], we establish the asymptotic expansions of the variance of the stochastic process for both long-time and short-time, which sharply improves the main result in [Proc. Amer. Math. Soc. 149 (2021), 2067–2080] by removing their technical conditions on the regularly varying behaviours and reformulating the asymptotic expansion in a more natural form. By analysing a new noncommutative operation on a subclass of completely positive functions, we provide a new way to construct finitely many ultraslow subdiffusion processes that are rapidly slower than a given ultraslow kernel. Consequently, we show that for a given completely monotonic ultraslow kernel, there is an induced kernel whose corresponding mean square displacement is logarithmic. [ABSTRACT FROM AUTHOR]

For spectrally positive Lévy processes killed on exiting the half-line, existence of a quasistationary distribution is characterized by the exponential integrability of the exit time, the Laplace exponent and the non-negativity of the scale functions. It is proven that if there is a quasi-stationary distribution, there are necessarily infinitely many ones and the set of quasi-stationary distributions is characterized. A sufficient condition for the minimal quasi-stationary distribution to be the Yaglom limit is given. [ABSTRACT FROM AUTHOR]

We study the long-time behavior for continuous-time Markov processes on the state space R≥0 := [0, ∞), which arise as unique strong solutions to stochastic equations with jumps. We establish, under a global dissipativity condition combined with a comparison principle, exponential ergodicity in various Wasserstein distances on R≥0. Our main emphasis lies on the derivation of these estimates under minimal moment conditions to be imposed on the associated Lévy measures of the noises. We apply our method to continuous-state branching processes with immigration (shorted as CBI processes), to nonlinear CBI processes, and finally to CBI processes in Lévy random environments. [ABSTRACT FROM AUTHOR]

We consider two jointly stationary and ergodic random measures ξ and η on Rd with equal finite intensities, assuming ξ to be diffuse (non-atomic). An allocation is a random mapping taking Rd to Rd υ {∞} in a translation invariant way. We construct allocations transporting the diffuse ξ to arbitrary η, under the mild condition of existence of an 'auxiliary' point process which is needed only in the case when η is diffuse. When that condition does not hold we show by a counterexample that an allocation transporting ξ to η need not exist. [ABSTRACT FROM AUTHOR]

We introduce the notion of a-walk S(n) = a1X1 + ... + anXn, based on a sequence of positive numbers a = (a1; a2; ...) and a Rademacher sequence X1;X2; .... We study recurrence/transience (properly defined) of such walks for various sequences of a. In particular, we establish the classification in the cases where ak = [kβ], β > 0, as well as in the case ak = [logγ k] or ak = logγ k for > 1. [ABSTRACT FROM AUTHOR]

We consider two jointly stationary and ergodic random measures ξ and η on Rd with equal finite intensities, assuming ξ to be diffuse (non-atomic). An allocation is a random mapping taking Rd to Rd υ {∞} in a translation invariant way. We construct allocations transporting the diffuse ξ to arbitrary η, under the mild condition of existence of an 'auxiliary' point process which is needed only in the case when η is diffuse. When that condition does not hold we show by a counterexample that an allocation transporting ξ to η need not exist. [ABSTRACT FROM AUTHOR]

A method for the analysis of super-resolution microscopy images is presented. This method is based on the analysis of stochastic trajectories of particles moving on the membrane of a cell with the assumption that this motion is determined by the properties of this membrane. Thus, the purpose of this method is to recover the structural properties of the membrane by solving an inverse problem governed by the Fokker–Planck equation related to the stochastic trajectories. Results of numerical experiments demonstrate the ability of the proposed method to reconstruct the potential of a cell membrane by using synthetic data similar those captured by super-resolution microscopy of luminescent activated proteins.