A star is born: Explosive Crump–Mode–Jagers branching processes

Controlled Branching Processes

In this paper we study blow up of the equation\(u_t = u_{xx} + u^\gamma \dot W_{tx}\), where\(\dot W_{tx}\) is a two-dimensional white noise field and where Dirichlet boundary conditions are enforced. It is known that if γ<3/2, then the solution exists for all time; in this paper we show that if γ is much larger than 3/2, then the solution blows up in finite time with positive probability. We prove this by considering how peaks in the solution propagate. If a peak of high mass forms, we rescale the equation and divide the mass of the peak into a collection of peaks of smaller mass, and these peaks evolve almost independently. In this way we compare the evolution ofu to a branching process. Large peaks are regarded as particles in this branching process. Offspring are peaks which are higher by some factor. We show that the expected number of offspring is greater than one when γ is much larger than 3/2, and thus the branching process survives with positive probability, corresponding to blowup in finite time.

Let $\{\mathbf{Z}_n\}$ be a multitype branching process in a random environment (MBPRE) which grows to infinity with positive probability for almost all environmental sequences. Under some conditions involving the first two moments of the environmental sequence, it is shown that dividing the $\{\mathbf{Z}_n\}$ components by their environment-conditioned expectations yields a sequence convergent in $L^2$ to a random vector with equal components.

Branching processes in a random environment are natural generalisations of Galton-Watson processes. In this paper we analyse the asymptotic decay of the survival probability for a sequence of slightly supercritical branching processes in an iid random environment, where the offspring expectation converges from above to 1. We prove that Haldane’s asymptotics, known from classical Galton-Watson processes, turns up again in the random environment case, provided that one stays away from the critical/subcritical regime. A central building block is a connection to and a limit theorem for perpetuities with asymptotically vanishing interest rates.

This review paper presents the known results on the asymptotics of the survival probability and limit theorems conditioned on survival of critical and subcritical branching processes in independent and identically distributed random environments. This is a natural generalization of the time-inhomogeneous branching processes. The key assumptions of the family of population models in question are nonoverlapping generations and discrete time. The reader should be aware of the fact that there are many very interesting papers covering other issues in the theory of branching processes in random environments which are not mentioned here.

We establish a general sufficient condition for a sequence of Galton–Watson branching processes in varying environments to converge weakly. This condition extends previ- ous results by allowing offspring distributions to have infinite variance. Our assumptions are stated in terms of pointwise convergence of a triplet of two real- valued functions and a measure. The limiting process is characterized by a backwards integro-differential equation satisfied by its Laplace exponent, which generalizes the branching equation satisfied by continuous state branching processes. Several examples are discussed, namely branching processes in random environment, Feller diffusion in varying environments and branching processes with catastrophes.

A branching process $Z(n)$, $n=0,1,\ldots,$ is considered which evolves in a random environment generated by a sequence of independent identically distributed generating functions $f_0(s),f_{1}(s),\ldots\;$. Let $S_0=0$, $S_k=\log f_0^{\prime}(1)+\cdots +\log f_{k-1}^{\prime }(1)$, $k\ge 1$, be the associated random walk and let $\tau (n)$ be the leftmost point of minimum of $\{S_k\}_{k\ge 0}$ on the interval $[0,n]$. Assuming that the random walk satisfies the Spitzer condition $n^{-1}\sum_{k=1}^{n}{\bf P}\{S_k>0\}\to\rho\in(0,1)$, $n\to \infty$, we show (under the quenched approach) that for each fixed $t\in (0,1]$ and $m=0,\pm 1,\pm 2,\ldots,$ the distribution of $Z(\tau (nt)+m)$ given $Z(n)>0$ converges as $n\to \infty$ to a (random) discrete distribution. Thus, in contrast to fixed points of the form $nt$, where the size of the population is large (even exponentially large; see [V. A. Vatutin and E. E. Dyakonova, Theory Probab. Appl., 49 (2005), pp. 275–309]), the size of the population at (random) p...

Let Zn be the number of particles at time n in a critical branching process in random environment and Zm,n be the number of particles in this process at time m ≤ n which have non-empty offspring at time n. We prove limit theorems for the process {Z[nt],n, t ∈ (0, 1]} conditioned on the event {Zn > 0}. Quenched and annealed approaches are considered.*

We study large deviation probabilities for bisexual branching process in a random (i.i.d.) environment. Under several conditions on the mating function (which may depend on the environment) we introduce the associated random walk of the process. We also assume Cramer condition for the step of the walk and moment conditions on the number of descendants of one pair. Under Cramer condition for the steps of the walk and moment conditions on the number of descendants of a pair we find an exact asymptotics of probabilities P(ln Nn ∈ [x, x + Δn )) as n → ∞ for x/n varying in some domain for all sequences Δn , tending to zero sufficiently slowly. Similar results we obtain for bisexual branching process with immigration in a random environment.

This research focuses on biological systems with sexual reproduction in which female and male individuals coexist together, forming female–male couples with the purpose of procreation. The couples can originate new females and males according to a certain probability law. Consequently, in this type of biological systems, two biological phases are involved: a mating phase in which the couples are formed, and a reproduction phase in which the couples, independently of the others, originate new offspring of both sexes. Due to several environmental factors of a random nature, these phases usually develop over time in a non-predictable (random) environment, frequently influenced by the numbers of females and males in the population and by the number of couples participating in the reproduction phase. In order to investigate the probabilistic evolution of these biological systems, in previous papers, by using a methodology based on branching processes, we had introduced a new class of two-sex mathematical models. Some probabilistic properties and limiting results were then established. Additionally, under a non-parametric statistical framework, namely, not assuming to have known the functional form of the offspring law, estimates for the main parameters affecting the reproduction phase were determined. We now continue this research line focusing the attention on the estimation of such reproductive parameters under a parametric statistical setting. In fact, we consider offspring probability laws belonging to the family of bivariate power series distributions. This general family includes the main probability distributions used to describe the offspring dynamic in biological populations with sexual reproduction. Under this parametric context, we propose accurate estimates for the parameters involved in the reproduction phase. With the aim of assessing the quality of the proposed estimates, we also determined optimal credibility intervals. For these purposes, we apply the Bayesian estimation methodology. As an illustration of the methodology developed, we present a simulated study about the demographic dynamics of Labord’s chameleon populations, where a sensitivity analysis on the prior density is included.

Propriétés asymptotiques des processus de branchement en environnement aléatoire

This paper examines the pricing of barrier options when the price of the underlying asset is modeled by a branching process in a random environment (BPRE). We derive an analytical formula for the price of an up-and-out call option, one form of a barrier option. Calibration of the model parameters is performed using market prices of standard call options. Our results show that the prices of barrier options that are priced with the BPRE model deviate significantly from those modeled assuming a lognormal process, despite the fact that for standard options, the corresponding differences between the two models are relatively small.

Two closely related results are established. A given critical branching process in a random environment attains a high level and spends at that level a part of its life obeying the arcsine law. If a critical Galton–Watson process survives up to a distant moment, then the ratio of the total number of individuals of the future generations to the total number of individuals ever born in the process obeys the arcsine law.

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Mobility models that have been used in the past to study delay tolerant networks (DTNs) have been either too complex to allow for deriving analytical expressions for performance measures, or have been too simplistic. In this paper we identify several classes of DTNs where the dynamics of the number of nodes that have a copy of some packet can be modeled as branching process with migration. Using recent results on such processes in a random environment, we obtain explicit formulae for the first two moments of the number of copies of a file that is propagated in the DTN, for quite general mobility models. Numerical examples illustrate our approach.