Let $X_t^\sharp$ be a multivariate process of the form $X_t =Y_t - Z_t$ , $X_0=x$ , killed at some terminal time T, where $Y_t$ is a Markov process having only jumps of length smaller than $\delta$ , and $Z_t$ is a compound Poisson process with jumps of length bigger than $\delta$ , for some fixed $\delta>0$ . Under the assumptions that the summands in $Z_t$ are subexponential, we investigate the asymptotic behaviour of the potential function $u(x)= \mathbb{E}^x \int_0^\infty \ell\big(X_s^\sharp\big)ds$ . The case of heavy-tailed entries in $Z_t$ corresponds to the case of ‘big claims’ in insurance models and is of practical interest. The main approach is based on the fact that u(x) satisfies a certain renewal equation.

The practically important classes of equal-input and of monotone Markov matrices are revisited, with special focus on embeddability, infinite divisibility, and mutual relations. Several uniqueness results for the classic Markov embedding problem are obtained in the process. To achieve our results, we need to employ various algebraic and geometric tools, including commutativity, permutation invariance, and convexity. Of particular relevance in several demarcation results are Markov matrices that are idempotents.

A continuous-state branching process with immigration having branching mechanism $\Psi$ and immigration mechanism $\Phi$ , a CBI $(\Psi,\Phi)$ process for short, may have either of two different asymptotic regimes, depending on whether $\int_{0}\frac{\Phi(u)}{|\Psi(u)|}\textrm{d} u<\infty$ or $\int_{0}\frac{\Phi(u)}{|\Psi(u)|}\textrm{d} u=\infty$ . When $\int_{0}\frac{\Phi(u)}{|\Psi(u)|}\textrm{d} u<\infty$ , the CBI process has either a limit distribution or a growth rate dictated by the branching dynamics. When $\scriptstyle\int_{0}\tfrac{\Phi(u)}{|\Psi(u)|}\textrm{d} u=\infty$ , immigration overwhelms branching dynamics. Asymptotics in the latter case are studied via a nonlinear time-dependent renormalization in law. Three regimes of weak convergence are exhibited. Processes with critical branching mechanisms subject to a regular variation assumption are studied. This article proves and extends results stated by M. Pinsky in ‘Limit theorems for continuous state branching processes with immigration’ (Bull. Amer. Math. Soc.78, 1972).

We correct typographical errors in our original paper.

The generalized perturbative approach is an all-purpose variant of Stein’s method used to obtain rates of normal approximation. Originally developed for functions of independent random variables, this method is here extended to functions of the realization of a hidden Markov model. In this dependent setting, rates of convergence are provided in some applications, leading, in each instance, to an extra log-factor vis-à-vis the rate in the independent case.

We solve non-Markovian optimal switching problems in discrete time on an infinite horizon, when the decision-maker is risk-aware and the filtration is general, and establish existence and uniqueness of solutions for the associated reflected backward stochastic difference equations. An example application to hydropower planning is provided.

In this paper, we construct operator fractional Lévy motion (ofLm), a broad class of infinitely divisible stochastic processes that are covariance operator self-similar and have wide-sense stationary increments. The ofLm class generalizes the univariate fractional Lévy motion as well as the multivariate operator fractional Brownian motion (ofBm). OfLm can be divided into two types, namely, moving average (maofLm) and real harmonizable (rhofLm), both of which share the covariance structure of ofBm under assumptions. We show that maofLm and rhofLm admit stochastic integral representations in the time and Fourier domains, and establish their distinct small- and large-scale limiting behavior. We also characterize time-reversibility for ofLm through parametric conditions related to its Lévy measure. In particular, we show that, under non-Gaussianity, the parametric conditions for time-reversibility are generally more restrictive than those for the Gaussian case (ofBm).

We study an N-player game where a pure action of each player is to select a nonnegative function on a Polish space supporting a finite diffuse measure, subject to a finite constraint on the integral of the function. This function is used to define the intensity of a Poisson point process on the Polish space. The processes are independent over the players, and the value to a player is the measure of the union of her open Voronoi cells in the superposition point process. Under randomized strategies, the process of points of a player is thus a Cox process, and the nature of competition between the players is akin to that in Hotelling competition games. We characterize when such a game admits Nash equilibria and prove that when a Nash equilibrium exists, it is unique and consists of pure strategies that are proportional in the same proportions as the total intensities. We give examples of such games where Nash equilibria do not exist. A better understanding of the criterion for the existence of Nash equilibria remains an intriguing open problem.

A moment constraint that limits the number of dividends in an optimal dividend problem is suggested. This leads to a new type of time-inconsistent stochastic impulse control problem. First, the optimal solution in the precommitment sense is derived. Second, the problem is formulated as an intrapersonal sequential dynamic game in line with Strotz’s consistent planning. In particular, the notions of pure dividend strategies and a (strong) subgame-perfect Nash equilibrium are adapted. An equilibrium is derived using a smooth fit condition. The equilibrium is shown to be strong. The uncontrolled state process is a fairly general diffusion.

Across a wide variety of applications, the self-exciting Hawkes process has been used to model phenomena in which the history of events influences future occurrences. However, there may be many situations in which the past events only influence the future as long as they remain active. For example, a person spreads a contagious disease only as long as they are contagious. In this paper, we define a novel generalization of the Hawkes process that we call the ephemerally self-exciting process. In this new stochastic process, the excitement from one arrival lasts for a randomly drawn activity duration, hence the ephemerality. Our study includes exploration of the process itself as well as connections to well-known stochastic models such as branching processes, random walks, epidemics, preferential attachment, and Bayesian mixture models. Furthermore, we prove a batch scaling construction of general, marked Hawkes processes from a general ephemerally self-exciting model, and this novel limit theorem both provides insight into the Hawkes process and motivates the model contained herein as an attractive self-exciting process in its own right.

For the gambler’s ruin problem with two players starting with the same amount of money, we show the playing time is stochastically maximized when the games are fair.

This paper studies a composite problem involving decision-making about the optimal entry time and dynamic consumption afterwards. In Stage 1, the investor has access to full market information subject to some information costs and needs to choose an optimal stopping time to initiate Stage 2; in Stage 2, the investor terminates the costly full information acquisition and starts dynamic investment and consumption under partial observation of free public stock prices. Habit formation preferences are employed, in which past consumption affects the investor’s current decisions. Using the stochastic Perron method, the value function of the composite problem is proved to be the unique viscosity solution of some variational inequalities.

Let $(Z_n)_{n\geq 0}$ be a critical branching process in a random environment defined by a Markov chain $(X_n)_{n\geq 0}$ with values in a finite state space $\mathbb{X}$ . Let $ S_n = \sum_{k=1}^n \ln f_{X_k}^{\prime}(1)$ be the Markov walk associated to $(X_n)_{n\geq 0}$ , where $f_i$ is the offspring generating function when the environment is $i \in \mathbb{X}$ . Conditioned on the event $\{ Z_n>0\}$ , we show the nondegeneracy of the limit law of the normalized number of particles ${Z_n}/{e^{S_n}}$ and determine the limit of the law of $\frac{S_n}{\sqrt{n}} $ jointly with $X_n$ . Based on these results we establish a Yaglom-type theorem which specifies the limit of the joint law of $ \log Z_n$ and $X_n$ given $Z_n>0$ .

This paper discusses a general class of replicator–mutator equations on a multidimensional fitness space. We establish a novel probabilistic representation of weak solutions of the equation by using the theory of Fokker–Planck–Kolmogorov (FPK) equations and a martingale extraction approach. We provide examples with closed-form probabilistic solutions for different fitness functions considered in the existing literature. We also construct a particle system and prove a general convergence result to the unique solution of the FPK equation associated with the extended replicator–mutator equation with respect to a Wasserstein-like metric adapted to our probabilistic framework.

Signal-to-interference-plus-noise ratio (SINR) percolation is an infinite-range dependent variant of continuum percolation modeling connections in a telecommunication network. Unlike in earlier works, in the present paper the transmitted signal powers of the devices of the network are assumed random, independent and identically distributed, and possibly unbounded. Additionally, we assume that the devices form a stationary Cox point process, i.e., a Poisson point process with stationary random intensity measure, in two or more dimensions. We present the following main results. First, under suitable moment conditions on the signal powers and the intensity measure, there is percolation in the SINR graph given that the device density is high and interferences are sufficiently reduced, but not vanishing. Second, if the interference cancellation factor $\gamma$ and the SINR threshold $\tau$ satisfy $\gamma \geq 1/(2\tau)$ , then there is no percolation for any intensity parameter. Third, in the case of a Poisson point process with constant powers, for any intensity parameter that is supercritical for the underlying Gilbert graph, the SINR graph also percolates with some small but positive interference cancellation factor.

We revisit the so-called cat-and-mouse Markov chain, studied earlier by Litvak and Robert (2012). This is a two-dimensional Markov chain on the lattice $\mathbb{Z}^2$ , where the first component (the cat) is a simple random walk and the second component (the mouse) changes when the components meet. We obtain new results for two generalisations of the model. First, in the two-dimensional case we consider far more general jump distributions for the components and obtain a scaling limit for the second component. When we let the first component be a simple random walk again, we further generalise the jump distribution of the second component. Secondly, we consider chains of three and more dimensions, where we investigate structural properties of the model and find a limiting law for the last component.