In this paper we introduce two new classes of stationary random simplicial tessellations, the so-called $\beta$ - and $\beta^{\prime}$ -Delaunay tessellations. Their construction is based on a space–time paraboloid hull process and generalizes that of the classical Poisson–Delaunay tessellation. We explicitly identify the distribution of volume-power-weighted typical cells, establishing thereby a remarkable connection to the classes of $\beta$ - and $\beta^{\prime}$ -polytopes. These representations are used to determine the principal characteristics of such cells, including volume moments, expected angle sums, and cell intensities.

The rich-get-richer rule reinforces actions that have been frequently chosen in the past. What happens to the evolution of individuals’ inclinations to choose an action when agents interact? Interaction tends to homogenize, while each individual dynamics tends to reinforce its own position. Interacting stochastic systems of reinforced processes have recently been considered in many papers, in which the asymptotic behavior is proven to exhibit almost sure synchronization. In this paper we consider models where, even if interaction among agents is present, absence of synchronization may happen because of the choice of an individual nonlinear reinforcement. We show how these systems can naturally be considered as models for coordination games or technological or opinion dynamics.

This paper presents a new matrix-infinite-product-form (MIP-form) solution for the stationary distribution in upper block-Hessenberg Markov chains (UBH-MCs). The existing MIP-form solution (Masuyama, Queueing Systems 92, 2019, pp. 173–200) requires a certain parameter set that satisfies both a Foster–Lyapunov drift condition and a convergence condition. In contrast, the new MIP-form solution requires no such parameter sets and no other conditions. The new MIP-form solution also has ‘quasi-algorithmic constructibility’, which is a newly introduced feature of being constructed by iterating infinitely many times a recursive procedure of finite complexity per iteration. This feature is not found in the other existing solutions for the stationary distribution in general UBH-MCs.

In this paper, we consider an extended class of univariate and multivariate generalized Pólya processes and study its properties. In the generalized Pólya process considered in [8], each occurrence of an event increases the stochastic intensity of the counting process. In the extended class studied in this paper, on the contrary, it decreases the stochastic intensity of the process, which induces a kind of negative dependence in the increments in the disjoint time intervals. First, we define the extended class of generalized Pólya processes and derive some preliminary results which will be used in the remaining part of the paper. It is seen that the extended class of generalized Pólya processes can be viewed as generalized pure death processes, where the death rate depends on both the state and the time. Based on the preliminary results, the main properties of the multivariate extended generalized Pólya process and meaningful characterizations are obtained. Finally, possible applications to reliability modeling are briefly discussed.

This article examines large-time behaviour of finite-state mean-field interacting particle systems. Our first main result is a sharp estimate (in the exponential scale) of the time required for convergence of the empirical measure process of the N-particle system to its invariant measure; we show that when time is of the order $\exp\{N\Lambda\}$ for a suitable constant $\Lambda > 0$ , the process has mixed well and it is close to its invariant measure. We then obtain large-N asymptotics of the second-largest eigenvalue of the generator associated with the empirical measure process when it is reversible with respect to its invariant measure. We show that its absolute value scales as $\exp\{{-}N\Lambda\}$ . The main tools used in establishing our results are the large deviation properties of the empirical measure process from its large-N limit. As an application of the study of large-time behaviour, we also show convergence of the empirical measure of the system of particles to a global minimum of a certain ‘entropy’ function when particles are added over time in a controlled fashion. The controlled addition of particles is analogous to the cooling schedule associated with the search for a global minimum of a function using the simulated annealing algorithm.

In this article we consider a Monte-Carlo-based method to filter partially observed diffusions observed at regular and discrete times. Given access only to Euler discretizations of the diffusion process, we present a new procedure which can return online estimates of the filtering distribution with no time-discretization bias and finite variance. Our approach is based upon a novel double application of the randomization methods of Rhee and Glynn (Operat. Res.63, 2015) along with the multilevel particle filter (MLPF) approach of Jasra et al. (SIAM J. Numer. Anal.55, 2017). A numerical comparison of our new approach with the MLPF, on a single processor, shows that similar errors are possible for a mild increase in computational cost. However, the new method scales strongly to arbitrarily many processors.

We consider the spectral analysis of several examples of bilateral birth–death processes and compute explicitly the spectral matrix and the corresponding orthogonal polynomials. We also use the spectral representation to study some probabilistic properties of the processes, such as recurrence, the invariant distribution (if it exists), and the probability current.

We propose non-asymptotic controls of the cumulative distribution function $\mathbb{P}(|X_{t}|\ge \varepsilon)$ , for any $t>0$ , $\varepsilon>0$ and any Lévy process X such that its Lévy density is bounded from above by the density of an $\alpha$ -stable-type Lévy process in a neighborhood of the origin.

We establish sufficient conditions for differentiability of the expected cost collected over a discrete-time Markov chain until it enters a given set. The parameter with respect to which differentiability is analysed may simultaneously affect the Markov chain and the set defining the stopping criterion. The general statements on differentiability lead to unbiased gradient estimators.

Bifurcating Markov chains (BMCs) are Markov chains indexed by a full binary tree representing the evolution of a trait along a population where each individual has two children. We provide a central limit theorem for additive functionals of BMCs under $L^2$ -ergodic conditions with three different regimes. This completes the pointwise approach developed in a previous work. As an application, we study the elementary case of a symmetric bifurcating autoregressive process, which justifies the nontrivial hypothesis considered on the kernel transition of the BMCs. We illustrate in this example the phase transition observed in the fluctuations.

Empirical studies (e.g. Jiang et al. (2015) and Mislove et al. (2007)) show that online social networks have not only in- and out-degree distributions with Pareto-like tails, but also a high proportion of reciprocal edges. A classical directed preferential attachment (PA) model generates in- and out-degree distributions with power-law tails, but the theoretical properties of the reciprocity feature in this model have not yet been studied. We derive asymptotic results on the number of reciprocal edges between two fixed nodes, as well as the proportion of reciprocal edges in the entire PA network. We see that with certain choices of parameters, the proportion of reciprocal edges in a directed PA network is close to 0, which differs from the empirical observation. This points out one potential problem of fitting a classical PA model to a given network dataset with high reciprocity, and indicates that alternative models need to be considered.

In this paper we propose a general framework for modeling an insurance liability cash flow in continuous time, by generalizing the reduced-form framework for credit risk and life insurance. In particular, we assume a nontrivial dependence structure between the reference filtration and the insurance internal filtration. We apply these results for pricing and hedging non-life insurance liabilities in hybrid financial and insurance markets, while taking into account the role of inflation under the benchmarked risk-minimization approach. This framework offers at the same time a general and flexible structure, and an explicit and treatable pricing-hedging formula.

We study the integrated telegraph process $X_t$ under the assumption of general distribution for the random times between consecutive reversals of direction. Specifically, $X_t$ represents the position, at time t, of a particle moving U time units upwards with velocity c and D time units downwards with velocity $-c$ . The latter motions are repeated cyclically, according to independent alternating renewals. Explicit expressions for the probability law of $X_t$ are given in the following cases: (i) (U, D) gamma-distributed; (ii) U exponentially distributed and D gamma-distributed. For certain values of the parameters involved, the probability law of $X_t$ is provided in a closed form. Some expressions for the moment generating function of $X_t$ and its Laplace transform are also obtained. The latter allows us to prove the existence of a Kac-type condition under which the probability density function of the integrated telegraph process, with identically distributed gamma intertimes, converges to that of the standard Brownian motion.

Finally, we consider the square of $X_t$ and disclose its distribution function, specifying the expression for some choices of the distribution of (U, D).

We apply general moment identities for Poisson stochastic integrals with random integrands to the computation of the moments of Markovian growth–collapse processes. This extends existing formulas for mean and variance available in the literature to closed-form moment expressions of all orders. In comparison with other methods based on differential equations, our approach yields explicit summations in terms of the time parameter. We also treat the case of the associated embedded chain, and provide recursive codes in Maple and Mathematica for the computation of moments and cumulants of any order with arbitrary cut-off moment sequences and jump size functions.

We consider residue expansions for survival and density/mass functions of first-passage distributions in finite-state semi-Markov processes (SMPs) in continuous and integer time. Conditions are given which guarantee that the residue expansions for these functions have a dominant exponential/geometric term. The key condition assumes that the relevant states for first passage contain an irreducible class, thus ensuring the same sort of dominant exponential/geometric terms as one gets for phase-type distributions in Markov processes. Essentially, the presence of an irreducible class along with some other conditions ensures that the boundary singularity b for the moment generating function (MGF) of the first-passage-time distribution is a simple pole. In the continuous-time setting we prove that b is a dominant pole, in that the MGF has no other pole on the vertical line $\{\text{Re}(s)=b\}.$ In integer time we prove that b is dominant if all holding-time mass functions for the SMP are aperiodic and non-degenerate. The expansions and pole characterisations address first passage to a single new state or a subset of new states, and first return to the starting state. Numerical examples demonstrate that the residue expansions are considerably more accurate than saddlepoint approximations and can provide a substitute for exact computation above the 75th percentile.

We study an intertemporal consumption and portfolio choice problem under Knightian uncertainty in which agent’s preferences exhibit local intertemporal substitution. We also allow for market frictions in the sense that the pricing functional is nonlinear. We prove existence and uniqueness of the optimal consumption plan, and we derive a set of sufficient first-order conditions for optimality. With the help of a backward equation, we are able to determine the structure of optimal consumption plans. We obtain explicit solutions in a stationary setting in which the financial market has different risk premia for short and long positions.

We consider two-dimensional Lévy processes reflected to stay in the positive quadrant. Our focus is on the non-standard regime when the mean of the free process is negative but the reflection vectors point away from the origin, so that the reflected process escapes to infinity along one of the axes. Under rather general conditions, it is shown that such behaviour is certain and each component can dominate the other with positive probability for any given starting position. Additionally, we establish the corresponding invariance principle providing justification for the use of the reflected Brownian motion as an approximate model. Focusing on the probability that the first component dominates, we derive a kernel equation for the respective Laplace transform in the starting position. This is done for the compound Poisson model with negative exponential jumps and, by means of approximation, for the Brownian model. Both equations are solved via boundary value problem analysis, which also yields the domination probability when starting at the origin. Finally, certain asymptotic analysis and numerical results are presented.

Restless bandits are a class of sequential resource allocation problems concerned with allocating one or more resources among several alternative processes where the evolution of the processes depends on the resources allocated to them. Such models capture the fundamental trade-offs between exploration and exploitation. In 1988, Whittle developed an index heuristic for restless bandit problems which has emerged as a popular solution approach because of its simplicity and strong empirical performance. The Whittle index heuristic is applicable if the model satisfies a technical condition known as indexability. In this paper, we present two general sufficient conditions for indexability and identify simpler-to-verify refinements of these conditions. We then revisit a previously proposed algorithm called the adaptive greedy algorithm which is known to compute the Whittle index for a sub-class of restless bandits. We show that a generalization of the adaptive greedy algorithm computes the Whittle index for all indexable restless bandits. We present an efficient implementation of this algorithm which can compute the Whittle index of a restless bandit with K states in $\mathcal{O}\!\left(K^3\right)$ computations. Finally, we present a detailed numerical study which affirms the strong performance of the Whittle index heuristic.

The hyperbolic random geometric graph was introduced by Krioukov et al. (Phys. Rev. E82, 2010). Among many equivalent models for the hyperbolic space, we study the d-dimensional Poincaré ball ( $d\ge 2$ ), with a general connectivity radius. While many phase transitions are known for the expectation asymptotics of certain subgraph counts, very little is known about the second-order results. Two of the distinguishing characteristics of geometric graphs on the hyperbolic space are the presence of tree-like hierarchical structures and the power-law behaviour of the degree distribution. We aim to reveal such characteristics in detail by investigating the behaviour of sub-tree counts. We show multiple phase transitions for expectation and variance in the resulting hyperbolic geometric graph. In particular, the expectation and variance of the sub-tree counts exhibit an intricate dependence on the degree sequence of the tree under consideration. Additionally, unlike the thermodynamic regime of the Euclidean random geometric graph, the expectation and variance may exhibit different growth rates, which is indicative of power-law behaviour. Finally, we also prove a normal approximation for sub-tree counts using the Malliavin–Stein method of Last et al. (Prob. Theory Relat. Fields165, 2016), along with the Palm calculus for Poisson point processes.

The study of multivariate extremes is dominated by multivariate regular variation, although it is well known that this approach does not provide adequate distinction between random vectors whose components are not always simultaneously large. Various alternative dependence measures and representations have been proposed, with the most well-known being hidden regular variation and the conditional extreme value model. These varying depictions of extremal dependence arise through consideration of different parts of the multivariate domain, and particularly through exploring what happens when extremes of one variable may grow at different rates from other variables. Thus far, these alternative representations have come from distinct sources, and links between them are limited. In this work we elucidate many of the relevant connections through a geometrical approach. In particular, the shape of the limit set of scaled sample clouds in light-tailed margins is shown to provide a description of several different extremal dependence representations.