Based on a simple object, an i.i.d. sequence of positive integer-valued random variables {an}n∊ℤ, we introduce and study two random structures and their connections. First, a population dynamics, in which each individual is born at time n and dies at time n + an. This dynamics is that of a D/GI/∞ queue, with arrivals at integer times and service times given by {an}n∊ℤ. Second, the directed random graph Tf on ℤ generated by the random map f(n) = n + an. Assuming only that E [a0] < ∞ and P [a0 = 1] > 0, we show that, in steady state, the population dynamics is regenerative, with one individual alive at each regeneration epoch. We identify a unimodular structure in this dynamics. More precisely, Tf is a unimodular directed tree, in which f(n) is the parent of n. This tree has a unique bi-infinite path. Moreover, Tf splits the integers into two categories: ephemeral integers, with a finite number of descendants of all degrees, and successful integers, with an infinite number. Each regeneration epoch is a successful individual such that all integers less than it are its descendants of some order. Ephemeral, successful, and regeneration integers form stationary and mixing point processes on ℤ.

A family of generalized ageing intensity functions of univariate absolutely continuous lifetime random variables is introduced and studied. They allow the analysis and measurement of the ageing tendency from various points of view. Some of these generalized ageing intensities characterize families of distributions dependent on a single parameter, while others determine distributions uniquely. In particular, it is shown that the elasticity functions of various transformations of distributions that appear in lifetime analysis and reliability theory uniquely characterize the parent distribution. Moreover, the recognition of the shape of a properly chosen generalized ageing intensity estimate admits a simple identification of the data lifetime distribution.

We introduce and study the notion of sure profits via flash strategies, consisting of a high-frequency limit of buy-and-hold trading strategies. In a fully general setting, without imposing any semimartingale restriction, we prove that there are no sure profits via flash strategies if and only if asset prices do not exhibit predictable jumps. This result relies on the general theory of processes and provides the most general formulation of the well-known fact that, in an arbitrage-free financial market, asset prices (including dividends) should not exhibit jumps of a predictable direction or magnitude at predictable times. We furthermore show that any price process is always right-continuous in the absence of sure profits. Our results are robust under small transaction costs and imply that, under minimal assumptions, price changes occurring at scheduled dates should only be due to unanticipated information releases.

In this paper we derive several explicit results on one special sticky diffusion process which is constructed as a time-changed version of a diffusion with no sticky points. A theorem concerning the process-related Green operators defined on some nonnegative piecewise continuous functions is provided. Then, based on this theorem, we explore the distributional properties of the sticky diffusion. A financial application is presented where we compute the value of the European vanilla call option written on the underlying with sticky price dynamics.

A new network evolution model is introduced in this paper. The model is based on cooperations of N units. The units are the nodes of the network and the cooperations are indicated by directed links. At each evolution step N units cooperate, which formally means that they form a directed N-star subgraph. At each step either a new unit joins the network and it cooperates with N − 1 old units, or N old units cooperate. During the evolution both preferential attachment and uniform choice are applied. Asymptotic power law distributions are obtained both for in-degrees and for out-degrees.

For spectrally negative Lévy processes, we prove several fluctuation results involving a general draw-down time, which is a downward exit time from a dynamic level that depends on the running maximum of the process. In particular, we find expressions of the Laplace transforms for the two-sided exit problems involving the draw-down time. We also find the Laplace transforms for the hitting time and creeping time over the running-maximum related draw-down level, respectively, and obtain an expression for a draw-down associated potential measure. The results are expressed in terms of scale functions for the spectrally negative Lévy processes.

A large and sparse random graph with independent exponentially distributed link weights can be used to model the propagation of messages or diseases in a network with an unknown connectivity structure. In this article we study an extended setting where, in addition, the nodes of the graph are equipped with nonnegative random weights which are used to model the effect of boundary delays across paths in the network. Our main results provide approximative formulas for typical first passage times, typical flooding times, and maximum flooding times in the extended setting, over a time scale logarithmic with respect to the network size.

For a spectrally negative Lévy process X, killed according to a rate that is a function ω of its position, we complement the recent findings of [12] by analysing (in greater generality) the exit probability of the one-sided upwards passage problem. When ω is strictly positive, this problem is related to the determination of the Laplace transform of the first passage time upwards for X that has been time-changed by the inverse of the additive functional $$\int_0^ \cdot \omega ({X_u}){\kern 1pt} {\rm{d}}u$$. In particular, our findings thus shed extra light on related results concerning first passage times downwards (resp. upwards) of continuous-state branching processes (resp. spectrally negative positive self-similar Markov processes).

We study the asymptotic behaviour of a class of small-noise diffusions driven by fractional Brownian motion, with random starting points. Different scalings allow for different asymptotic properties of the process (small-time and tail behaviours in particular). In order to do so, we extend some results on sample path large deviations for such diffusions. As an application, we show how these results characterise the small-time and tail estimates of the implied volatility for rough volatility models, recently proposed in mathematical finance.

This paper considers two variants of M/G/1 queues with impatient customers, which are denoted by M/G/1+Gw and M/G/1+Gs. In the M/G/1+Gw queue customers have deadlines for their waiting times, and they leave the system immediately if their services do not start before the expiration of their deadlines. On the other hand, in the M/G/1+Gs queue customers have deadlines for their sojourn times, where customers in service also immediately leave the system when their deadlines expire. In this paper we derive comparison results for performance measures of these models. In particular, we show that if the service time distribution is new better than used in expectation, then the loss probability in the M/G/1+Gs queue is greater than that in the M/G/1+Gw queue.

The accessibility percolation model is investigated on random rooted labeled trees. More precisely, the number of accessible leaves (i.e. increasing paths) Zn and the number of accessible vertices Cn in a random rooted labeled tree of size n are jointly considered in this work. As n → ∞, we prove that (Zn, Cn) converges in distribution to a random vector whose probability generating function is given in an explicit form. In particular, we obtain that the asymptotic distributions of Zn + 1 and Cn are geometric distributions with parameters e/(1 + e) and 1/e, respectively. Much of our analysis is performed in the context of local weak convergence of random rooted labeled trees.

The De Vylder and Goovaerts conjecture is an open problem in risk theory, stating that the finite-time ruin probability in a standard risk model is greater than or equal to the corresponding ruin probability evaluated in an associated model with equalized claim amounts. Equalized means here that the jump sizes of the associated model are equal to the average jump in the initial model between 0 and a terminal time T.In this paper, we consider the diffusion approximations of both the standard risk model and its associated risk model. We prove that the associated model, when conveniently renormalized, converges in distribution to a Gaussian process satisfying a simple SDE. We then compute the probability that this diffusion hits the level 0 before time T and compare it with the same probability for the diffusion approximation for the standard risk model. We conclude that the De Vylder and Goovaerts conjecture holds for the diffusion limits.

Let (X, Y) = (Xn, Yn)n≥1 be the output process generated by a hidden chain Z = (Zn)n≥1, where Z is a finite-state, aperiodic, time homogeneous, and irreducible Markov chain. Let LCn be the length of the longest common subsequences of X1,..., Xn and Y1,..., Yn. Under a mixing hypothesis, a rate of convergence result is obtained for E[LCn]/n.

Let X1, …, Xn be independent random points drawn from an absolutely continuous probability measure with density f in ℝd. Under mild conditions on f, wederive a Poisson limit theorem for the number of large probability nearest-neighbour balls. Denoting by Pn the maximum probability measure of nearest-neighbour balls, this limit theorem implies a Gumbel extreme value distribution for nPn − ln n as n → ∞. Moreover, we derive a tight upper bound on the upper tail of the distribution of nPn − ln n, which does not depend on f.

Letting ℳ denote the space of finite measures on ℕ, and μλ ∊ ℳ denote the Poisson distribution with parameter λ, the function W : [0, 1]2 → ℳ given by W(x, y) = μc log x log y is called the PAG graphon with density c. It is known that this is the limit, in the multigraph homomorphism sense, of the dense preferential attachment graph (PAG) model with edge density c. This graphon can then in turn be used to generate the so-called W-random graphs in a natural way, and similar constructions also work in the slightly more general context of the so-called PAGκ models. The aim of this paper is to compare these dense PAGκ models with the W-random graph models obtained from the corresponding graphons. Motivated by the multigraph limit theory, we investigate the expected jumble-norm distance of the two models in terms of the number of vertices n. We present a coupling for which the expectation can be bounded from above by O(log3/2n · n−1/2), and provide a universal lower bound that is coupling-independent, but without the logarithmic term.

Martingales constitute a basic tool in stochastic analysis; this paper considers their application to counting processes. We use this tool to revisit a renewal theorem and give extensions for various counting processes. We first consider a renewal process as a pilot example, deriving a new semimartingale representation that differs from the standard decomposition via the stochastic intensity function. We then revisit Blackwell’s renewal theorem, its refinements and extensions. Based on these observations, we extend the semimartingale representation to a general counting process, and give conditions under which asymptotic behaviour similar to Blackwell’s renewal theorem holds.

We characterize various forms of positive dependence, such as association, positive supermodular association and dependence, and positive orthant dependence, for jump-Feller processes. Such jump processes can be studied through their state-space dependent Lévy measures. It is through these Lévy measures that we will provide our characterization. Finally, we present applications of these results to stochastically monotone Feller processes, including Lévy processes, the Ornstein–Uhlenbeck process, pseudo-Poisson processes, and subordinated Feller processes.