The distributed genome hypothesis states that the set of genes in a population of bacteria is distributed over all individuals that belong to the specific taxon. It implies that certain genes can be gained and lost from generation to generation. We use the random genealogy given by a Kingman coalescent in order to superimpose events of gene gain and loss along ancestral lines. Gene gains occur at a constant rate along ancestral lines. We assume that gained genes have never been present in the population before. Gene losses occur at a rate proportional to the number of genes present along the ancestral line. In this infinitely many genes model we derive moments for several statistics within a sample: the average number of genes per individual, the average number of genes differing between individuals, the number of incongruent pairs of genes, the total number of different genes in the sample and the gene frequency spectrum. We demonstrate that the model gives a reasonable fit with gene frequency data from marine cyanobacteria.

We propose a numerical method to approximate the value function for the optimal stopping problem of a piecewise deterministic Markov process (PDMP). Our approach is based on quantization of the post jump location—inter-arrival time Markov chain naturally embedded in the PDMP, and path-adapted time discretization grids. It allows us to derive bounds for the convergence rate of the algorithm and to provide a computable ε-optimal stopping time. The paper is illustrated by a numerical example.

In this paper, the forgetting of the initial distribution for a nonergodic Hidden Markov Models (HMM) is studied. A new set of conditions is proposed to establish the forgetting property of the filter. Both a pathwise and mean convergence of the total variation distance of the filter started from two different initial distributions are obtained. The results are illustrated using a generic nonergodic state-space model for which both pathwise and mean exponential stability is established.

We derive a Poisson random field model for population site polymorphisms differences within and between two species that share a relatively recent common ancestor. The model can be either equilibrium or time inhomogeneous. We first consider a random field of Markov chains that describes the fate of a set of individual mutations. This field is approximated by a Poisson random field from which we can make inferences about the amounts of mutation and selection that have occurred in the history of observed aligned DNA sequences.

We provide an axiomatic foundation for the representation of numéraire-invariant preferences of economic agents acting in a financial market. In a static environment, the simple axioms turn out to be equivalent to the following choice rule: the agent prefers one outcome over another if and only if the expected (under the agent’s subjective probability) relative rate of return of the latter outcome with respect to the former is nonpositive. With the addition of a transitivity requirement, this last preference relation has an extension that can be numerically represented by expected logarithmic utility. We also treat the case of a dynamic environment where consumption streams are the objects of choice. There, a novel result concerning a canonical representation of unit-mass optional measures enables us to explicitly solve the investment–consumption problem by separating the two aspects of investment and consumption. Finally, we give an application to the problem of optimal numéraire investment with a random time-horizon.

Multifractal analysis of multiplicative random cascades is revisited within the framework of mixed asymptotics. In this new framework, the observed process can be modeled by a concatenation of independent binary cascades and statistics are estimated over a sample whose size increases as the resolution scale (or the sampling period) becomes finer. This allows one to continuously interpolate between the situation where one studies a single cascade sample at arbitrary fine scales and where, at fixed scale, the sample length (number of cascades realizations) becomes infinite. We show that scaling exponents of “mixed” partitions functions, that is, the estimator of the cumulant generating function of the cascade generator distribution depends on some “mixed asymptotic” exponent χ, respectively, above and below two critical value p_χ⁻ and p_χ⁺. We study the convergence properties of partition functions in mixed asymtotics regime and establish a central limit theorem. Moreover, within the mixed asymptotic framework, we establish a “box-counting” multifractal formalism that can be seen as a rigorous formulation of Mandelbrot’s negative dimension theory. Numerical illustrations of our results on specific examples are also provided. A possible application of these results is to distinguish data generated by log-Normal or log-Poisson models.

We develop a McKean–Vlasov interpretation of Navier–Stokes equations with external force field in the whole space, by associating with local mild L^p-solutions of the 3d-vortex equation a generalized nonlinear diffusion with random space–time birth that probabilistically describes creation of rotation in the fluid due to nonconservativeness of the force. We establish a local well-posedness result for this process and a stochastic representation formula for the vorticity in terms of a vector-weighted version of its law after its birth instant. Then we introduce a stochastic system of 3d vortices with mollified interaction and random space–time births, and prove the propagation of chaos property, with the nonlinear process as limit, at an explicit pathwise convergence rate. Convergence rates for stochastic approximation schemes of the velocity and the vorticity fields are also obtained. We thus extend and refine previous results on the probabilistic interpretation and stochastic approximation methods for the nonforced equation, generalizing also a recently introduced random space–time-birth particle method for the 2d-Navier–Stokes equation with force.

In this paper we introduce a ten-parameter family of Lévy processes for which we obtain Wiener–Hopf factors and distribution of the supremum process in semi-explicit form. This family allows an arbitrary behavior of small jumps and includes processes similar to the generalized tempered stable, KoBoL and CGMY processes. Analytically it is characterized by the property that the characteristic exponent is a meromorphic function, expressed in terms of beta and digamma functions. We prove that the Wiener–Hopf factors can be expressed as infinite products over roots of a certain transcendental equation, and the density of the supremum process can be computed as an exponentially converging infinite series. In several special cases when the roots can be found analytically, we are able to identify the Wiener–Hopf factors and distribution of the supremum in closed form. In the general case we prove that all the roots are real and simple, and we provide localization results and asymptotic formulas which allow an efficient numerical evaluation. We also derive a convergence acceleration algorithm for infinite products and a simple and efficient procedure to compute the Wiener–Hopf factors for complex values of parameters. As a numerical example we discuss computation of the density of the supremum process.

Starting from a given Markov kernel on a finite set V and a bijection g of V, we construct and study a time inhomogeneous Markov chain whose kernel at time n is obtained from K by transport of gⁿ⁻¹. We show that this construction leads to interesting examples, and we obtain quantitative results for some of these examples.

This paper studies many-server limits for multi-server queues that have a phase-type service time distribution and allow for customer abandonment. The first set of limit theorems is for critically loaded G/Ph/n+GI queues, where the patience times are independent and identically distributed following a general distribution. The next limit theorem is for overloaded G/Ph/n+M queues, where the patience time distribution is restricted to be exponential. We prove that a pair of diffusion-scaled total-customer-count and server-allocation processes, properly centered, converges in distribution to a continuous Markov process as the number of servers n goes to infinity. In the overloaded case, the limit is a multi-dimensional diffusion process, and in the critically loaded case, the limit is a simple transformation of a diffusion process. When the queues are critically loaded, our diffusion limit generalizes the result by Puhalskii and Reiman (2000) for GI/Ph/n queues without customer abandonment. When the queues are overloaded, the diffusion limit provides a refinement to a fluid limit and it generalizes a result by Whitt (2004) for M/M/n/+M queues with an exponential service time distribution. The proof techniques employed in this paper are innovative. First, a perturbed system is shown to be equivalent to the original system. Next, two maps are employed in both fluid and diffusion scalings. These maps allow one to prove the limit theorems by applying the standard continuous-mapping theorem and the standard random-time-change theorem.

In this paper we consider Dynkin’s games with payoffs which are functions of an underlying process. Assuming extended weak convergence of underlying processes {S⁽ⁿ⁾}_n=0^∞ to a limit process S we prove convergence Dynkin’s games values corresponding to {S⁽ⁿ⁾}_n=0^∞ to the Dynkin’s game value corresponding to S. We use these results to approximate game options prices with path dependent payoffs in continuous time models by a sequence of game options prices in discrete time models which can be calculated by dynamical programming algorithms. In comparison to previous papers we work under more general convergence of underlying processes, as well as weaker conditions on the payoffs.

We study first passage percolation on the configuration model. Assuming that each edge has an independent exponentially distributed edge weight, we derive explicit distributional asymptotics for the minimum weight between two randomly chosen connected vertices in the network, as well as for the number of edges on the least weight path, the so-called hopcount.

We analyze the configuration model with degree power-law exponent τ>2, in which the degrees are assumed to be i.i.d. with a tail distribution which is either of power-law form with exponent τ−1>1, or has even thinner tails (τ=∞). In this model, the degrees have a finite first moment, while the variance is finite for τ>3, but infinite for τ∈(2, 3).

We prove a central limit theorem for the hopcount, with asymptotically equal means and variances equal to αlogn, where α∈(0, 1) for τ∈(2, 3), while α>1 for τ>3. Here n denotes the size of the graph. For τ∈(2, 3), it is known that the graph distance between two randomly chosen connected vertices is proportional to loglogn [Electron. J. Probab. 12 (2007) 703–766], that is, distances are ultra small. Thus, the addition of edge weights causes a marked change in the geometry of the network. We further study the weight of the least weight path and prove convergence in distribution of an appropriately centered version.

This study continues the program initiated in [J. Math. Phys. 49 (2008) 125218] of showing that logn is the correct scaling for the hopcount under i.i.d. edge disorder, even if the graph distance between two randomly chosen vertices is of much smaller order. The case of infinite mean degrees (τ∈[1, 2)) is studied in [Extreme value theory, Poisson–Dirichlet distributions and first passage percolation on random networks (2009) Preprint] where it is proved that the hopcount remains uniformly bounded and converges in distribution.