The existence of moments of first downward passage times of a spectrally negative Lévy process is governed by the general dynamics of the Lévy process, i.e. whether it is drifting to $+\infty$ , $-\infty$ , or oscillating. Whenever the Lévy process drifts to $+\infty$ , we prove that the $\kappa$ th moment of the first passage time (conditioned to be finite) exists if and only if the $(\kappa+1)$ th moment of the Lévy jump measure exists. This generalizes a result shown earlier by Delbaen for Cramér–Lundberg risk processes. Whenever the Lévy process drifts to $-\infty$ , we prove that all moments of the first passage time exist, while for an oscillating Lévy process we derive conditions for non-existence of the moments, and in particular we show that no integer moments exist.

We analyse an additive-increase and multiplicative-decrease (also known as growth–collapse) process that grows linearly in time and that, at Poisson epochs, experiences downward jumps that are (deterministically) proportional to its present position. For this process, and also for its reflected versions, we consider one- and two-sided exit problems that concern the identification of the laws of exit times from fixed intervals and half-lines. All proofs are based on a unified first-step analysis approach at the first jump epoch, which allows us to give explicit, yet involved, formulas for their Laplace transforms. All eight Laplace transforms can be described in terms of two so-called scale functions associated with the upward one-sided exit time and with the upward two-sided exit time. All other Laplace transforms can be obtained from the above scale functions by taking limits, derivatives, integrals, and combinations of these.

Pseudo cross-variograms appear naturally in the context of multivariate Brown–Resnick processes, and are a useful tool for analysis and prediction of multivariate random fields. We give a necessary and sufficient criterion for a matrix-valued function to be a pseudo cross-variogram, and further provide a Schoenberg-type result connecting pseudo cross-variograms and multivariate correlation functions. By means of these characterizations, we provide extensions of the popular univariate space–time covariance model of Gneiting to the multivariate case.

We prove some estimates for the variations of transition probabilities of the (1+1)-affine process. From these estimates we deduce the strong Feller and the ergodic properties of the total variation distance of the process. The key strategy is the pathwise construction and analysis of several Markov couplings using strong solutions of stochastic equations.

We derive a new theoretical lower bound for the expected supremum of drifted fractional Brownian motion with Hurst index $H\in(0,1)$ over a (in)finite time horizon. Extensive simulation experiments indicate that our lower bound outperforms the Monte Carlo estimates based on very dense grids for $H\in(0,\tfrac{1}{2})$. Additionally, we derive the Paley–Wiener–Zygmund representation of a linear fractional Brownian motion in the general case and give an explicit expression for the derivative of the expected supremum at $H=\tfrac{1}{2}$ in the sense of Bisewski, Dȩbicki and Rolski (2021).

We examine urn models under random replacement schemes, and the related distributions, by using generating functions. A fundamental isomorphism between urn models and a certain system of differential equations has previously been established. We study the joint distribution of the numbers of balls in the urn, and determined recurrence relations for the probability generating functions. The associated partial differential equation satisfied by the generating function is derived. We develop analytical methods for the study of urn models that can lead to perspectives on urn-related problems from analytic combinatorics. The results presented here provide a broader framework for the study of exactly solvable urns than the existing framework. Finally, we examine several applications and their numerical results in order to demonstrate how our theoretical results can be employed in the study of urn models.

We consider an (R, Q) inventory model with two types of orders, normal orders and emergency orders, which are issued at different inventory levels. These orders are delivered after exponentially distributed lead times. In between deliveries, the inventory level decreases in a state-dependent way, according to a release rate function $\alpha({\cdot})$ . This function represents the fluid demand rate; it could be controlled by a system manager via price adaptations. We determine the mean number of downcrossings $\theta(x)$ of any level x in one regenerative cycle, and use it to obtain the steady-state density f (x) of the inventory level. We also derive the rates of occurrence of normal deliveries and of emergency deliveries, and the steady-state probability of having zero inventory.

We establish exponential ergodicity for a class of Markov processes with interactions, including two-factor type processes and Gruschin type processes. The proof is elementary and direct via the Markov coupling technique.

We present a reflection principle for a wide class of symmetric random motions with finite velocities. We propose a deterministic argument which is then applied to trajectories of stochastic processes. In the case of symmetric correlated random walks and the symmetric telegraph process, we provide a probabilistic result recalling the classical reflection principle for Brownian motion, but where the initial velocity has a crucial role. In the case of the telegraph process we also present some consequences which lead to further reflection-type characteristics of the motion.

Stein’s method is used to study discrete representations of multidimensional distributions that arise as approximations of states of quantum harmonic oscillators. These representations model how quantum effects result from the interaction of finitely many classical ‘worlds’, with the role of sample size played by the number of worlds. Each approximation arises as the ground state of a Hamiltonian involving a particular interworld potential function. Our approach, framed in terms of spherical coordinates, provides the rate of convergence of the discrete approximation to the ground state in terms of Wasserstein distance. Applying a novel Stein’s method technique to the radial component of the ground state solution, the fastest rate of convergence to the ground state is found to occur in three dimensions.

We consider a one-dimensional superprocess with a supercritical local branching mechanism $\psi$, where particles move as a Brownian motion with drift $-\rho$ and are killed when they reach the origin. It is known that the process survives with positive probability if and only if $\rho<\sqrt{2\alpha}$, where $\alpha=-\psi'(0)$. When $\rho<\sqrt{2 \alpha}$, Kyprianou et al. [18] proved that $\lim_{t\to \infty}R_t/t =\sqrt{2\alpha}-\rho$ almost surely on the survival set, where $R_t$ is the rightmost position of the support at time t. Motivated by this work, we investigate its large deviation, in other words, the convergence rate of $\mathbb{P}_{\delta_x} (R_t >\gamma t+\theta)$ as $t \to \infty$, where $\gamma >\sqrt{2 \alpha} -\rho$, $\theta \ge 0$. As a by-product, a related Yaglom-type conditional limit theorem is obtained. Analogous results for branching Brownian motion can be found in Harris et al. [13].

Considering a representative agent in the market, we study the long-term optimal investment problem in a discrete-time financial market, introducing a set of restrictions in the admissible strategies. The drawdown constraints limit the size of possible losses of the portfolio and impose a floor-based performance measure. The optimal growth rate is characterized, and under suitable hypotheses it is proved that an optimal strategy exists. The approach to solving this problem is based on dynamic programming techniques and a fixed point argument adapted from the theory of Markov decision processes.

This paper addresses the asymptotic analysis of sojourn functionals of spatiotemporal Gaussian random fields with long-range dependence (LRD) in time, also known as long memory. Specifically, reduction theorems are derived for local functionals of nonlinear transformation of such fields, with Hermite rank $m\geq 1,$ under general covariance structures. These results are proven to hold, in particular, for a family of nonseparable covariance structures belonging to the Gneiting class. For $m=2,$ under separability of the spatiotemporal covariance function in space and time, the properly normalized Minkowski functional, involving the modulus of a Gaussian random field, converges in distribution to the Rosenblatt-type limiting distribution for a suitable range of values of the long-memory parameter.

We study an optimal reinsurance problem for a diffusion model, in which the drift of the claim follows an Ornstein–Uhlenbeck process. The aim of the insurer is to maximize the expected exponential utility of its terminal wealth. We consider two cases: full information and partial information. Full information occurs when the insurer directly observes the drift; partial information occurs when the insurer observes only its claims. By applying stochastic control and by solving the corresponding Hamilton–Jacobi–Bellman equations, we find the value function and the optimal reinsurance strategy under both full and partial information. We determine a relationship between the value function and reinsurance strategy under full information with the value function and reinsurance strategy under partial information.

Growth-fragmentation processes describe the evolution of systems in which cells grow slowly and fragment suddenly. Despite originating as a way to describe biological phenomena, they have recently been found to describe the lengths of certain curves in statistical physics models. In this note, we describe a new growth-fragmentation process connected to random planar maps with faces of large degree, having as a key ingredient the ricocheted stable process recently discovered by Budd. The process has applications to the excursions of planar Brownian motion and Liouville quantum gravity.

Large deviations of the largest and smallest eigenvalues of $\mathbf{X}\mathbf{X}^\top/n$ are studied in this note, where $\mathbf{X}_{p\times n}$ is a $p\times n$ random matrix with independent and identically distributed (i.i.d.) sub-Gaussian entries. The assumption imposed on the dimension size p and the sample size n is $p=p(n)\rightarrow\infty$ with $p(n)={\mathrm{o}}(n)$. This study generalizes one result obtained in [3].

Let $(Z_n)_{n\geq0}$ be a supercritical Galton–Watson process. Consider the Lotka–Nagaev estimator for the offspring mean. In this paper we establish self-normalized Cramér-type moderate deviations and Berry–Esseen bounds for the Lotka–Nagaev estimator. The results are believed to be optimal or near-optimal.

We introduce a general two-colour interacting urn model on a finite directed graph, where each urn at a node reinforces all the urns in its out-neighbours according to a fixed, non-negative, and balanced reinforcement matrix. We show that the fraction of balls of either colour converges almost surely to a deterministic limit if either the reinforcement is not of Pólya type or the graph is such that every vertex with non-zero in-degree can be reached from some vertex with zero in-degree. We also obtain joint central limit theorems with appropriate scalings. Furthermore, in the remaining case when there are no vertices with zero in-degree and the reinforcement is of Pólya type, we restrict our analysis to a regular graph and show that the fraction of balls of either colour converges almost surely to a finite random limit, which is the same across all the urns.

We consider a class of processes describing a population consisting of k types of individuals. The process is almost surely absorbed at the origin within finite time, and we study the expected time taken for such extinction to occur. We derive simple and precise asymptotic estimates for this expected persistence time, starting either from a single individual or from a quasi-equilibrium state, in the limit as a system size parameter N tends to infinity. Our process need not be a Markov process on $ {\mathbb Z}_+^k$; we allow the possibility that individuals’ lifetimes may follow more general distributions than the exponential distribution.

Let $n\geq 2$ random lines intersect a planar convex domain D. Consider the probabilities $p_{nk}$ , $k=0,1, \ldots, {n(n-1)}/{2}$ that the lines produce exactly k intersection points inside D. The objective is finding $p_{nk}$ through geometric invariants of D. Using Ambartzumian’s combinatorial algorithm, the known results are instantly reestablished for $n=2, 3$ . When $n=4$ , these probabilities are expressed by new invariants of D. When D is a disc of radius r, the simplest forms of all invariants are found. The exact values of $p_{3k}$ and $p_{4k}$ are established.