Let $(X_t)_{t \geqslant 0}$ be the solution of the stochastic differential equation

\[\mathrm{d} X_t = b(X_t) \,\mathrm{d} t+A\,\mathrm{d} Z_t, \quad X_{0}=x,\]

$b\,:\, \mathbb{R}^d \rightarrow \mathbb{R}^d$

$A \in \mathbb{R}^{d \times d}$

$(Z_t)_{t\geqslant 0}$

$\alpha$

$\alpha \in (1,2)$

$x\in\mathbb{R}^{d}$

$\Gamma = (\gamma_n)_{n\in \mathbb{N}}$

$(X_t)_{t \geqslant 0}$

$\alpha$

$\gamma^{\frac{1}{\alpha}}_n$

$\gamma^{1+\frac {1}{\alpha}-\frac{1}{\kappa}}_n$

$\kappa \in [1,\alpha)$

$\gamma^{\frac{2-\alpha}{\alpha}}_n$

$\alpha$

Let X be the sum of a diffusion process and a Lévy jump process, and for any integer $n\ge 1$ let $\phi_n$ be a function defined on $\mathbb{R}^2$ and taking values in $\mathbb{R}$, with adequate properties. We study the convergence of functionals of the type

\begin{align*} \Delta_n\sum_{i=1}^{[t/\Delta_n]} \phi_n\!\left(\alpha_n\left[\frac{X_{(i-1)\Delta_n}}{\alpha_n}\right], \:\frac{\alpha_n}{\sqrt{\Delta_n}}\left(\left[\frac{X_{i\Delta_n}}{\alpha_n}\right]-\left[\frac{X_{(i-1)\Delta_n}}{\alpha_n}\right]\right)\right),\end{align*}

$(\Delta_n)$

$(\alpha_n)$

$n\to +\infty$

$\frac{\alpha_n}{\sqrt{\Delta_n}}$

$[0,+\infty)$]

We study an optimal inventory control problem under a reflected jump–diffusion netflow process with state-dependent jumps, in which the intensity of the jump process can depend on the inventory level. We examine the well-posedness of the associated integro-differential Hamilton–Jacobi–Bellman (ID-HJB) equation with Neumann boundary condition in the classical sense. To achieve this, we first establish the existence of viscosity solutions to the ID-HJB equation of an auxiliary control problem with a compact policy space, which is proved to be equivalent to the primal problem. We reformulate the ID-HJB equation as a Neumann HJB equation with the (non-local) integral term expressed in terms of the value function of the auxiliary problem and prove the existence of a unique classical solution to the Neumann HJB equation. Then, the well-posedness of the primal ID-HJB equation follows from the unique classical solution of the Neumann HJB equation and the existence of viscosity solutions to the auxiliary ID-HJB equation. Based on this classical solution, we characterize the optimal (admissible) inventory control strategy and show the verification result for the primal control problem.

This paper concerns an insurance firm’s surplus process observed at renewal inspection times, with a focus on assessing the probability of the surplus level dropping below zero. For various types of inter-inspection time distributions, an explicit expression for the corresponding transform is given. In addition, Cramér–Lundberg-type asymptotics are established. Also, an importance sampling-based Monte Carlo algorithm is proposed, and is shown to be logarithmically efficient.

Motivated by recent developments of quasi-stationary Monte Carlo methods, we investigate the stability of quasi-stationary distributions of killed Markov processes under perturbations of the generator. We first consider a general bounded self-adjoint perturbation operator, and then study a particular unbounded perturbation corresponding to truncation of the killing rate. In both scenarios, we quantify the difference between eigenfunctions of the smallest eigenvalue of the perturbed and unperturbed generators in a Hilbert space norm. As a consequence, $\mathcal{L}^1$-norm estimates of the difference of the resulting quasi-stationary distributions in terms of the perturbation are provided.

The preferential attachment model is a natural and popular random graph model for a growing network that contains very well-connected ‘hubs’. We study the higher-order connectivity of such a network by investigating the topological properties of its clique complex. We concentrate on the Betti numbers, a sequence of topological invariants of the complex related to the numbers of holes (equivalently, repeated connections) of different dimensions. We prove that the expected Betti numbers grow sublinearly fast, with the trivial exceptions of those at dimensions 0 and 1. Our result also shows that preferential attachment graphs undergo infinitely many phase transitions within the parameter regime where the limiting degree distribution has an infinite variance. Regarding higher-order connectivity, our result shows that preferential attachment favors higher-order connectivity. We illustrate our theoretical results with simulations.

This paper characterizes irreducible phase-type representations for exponential distributions. Bean and Green (2000) gave a set of necessary and sufficient conditions for a phase-type distribution with an irreducible generator matrix to be exponential. We extend these conditions to irreducible representations, and we thus give a characterization of all irreducible phase-type representations for exponential distributions. We consider the results in relation to time-reversal of phase-type distributions, PH-simplicity, and the algebraic degree of a phase-type distribution, and we give applications of the results. In particular we give the conditions under which a Coxian distribution becomes exponential, and we construct bivariate exponential distributions. Finally, we translate the main findings to the discrete case of geometric distributions.

For a continuous-time phase-type (PH) distribution, starting with its Laplace–Stieltjes transform, we obtain a necessary and sufficient condition for its minimal PH representation to have the same order as its algebraic degree. To facilitate finding this minimal representation, we transform this condition equivalently into a non-convex optimization problem, which can be effectively addressed using an alternating minimization algorithm. The algorithm convergence is also proved. Moreover, the method we develop for the continuous-time PH distributions can be used directly for the discrete-time PH distributions after establishing an equivalence between the minimal representation problems for continuous-time and discrete-time PH distributions.

We model voting behaviour in the multi-group setting of a two-tier voting system using sequences of de Finetti measures. Our model is defined by using the de Finetti representation of a probability measure (i.e. as a mixture of conditionally independent probability measures) describing voting behaviour. The de Finetti measure describes the interaction between voters and possible outside influences on them. We assume that for each population size there is a (potentially) different de Finetti measure, and as the population grows, the sequence of de Finetti measures converges weakly to the Dirac measure at the origin, representing a tendency toward weakening social cohesion as the population grows large. The resulting model covers a wide variety of behaviours, ranging from independent voting in the limit under fast convergence, a critical convergence speed with its own pattern of behaviour, to a subcritical convergence speed which yields a model in line with empirical evidence of real-world voting data, contrary to previous probabilistic models used in the study of voting. These models can be used, e.g., to study the problem of optimal voting weights in two-tier voting systems.

We introduce and study a game-theoretic model to understand the spread of an epidemic in a homogeneous population. A discrete-time stochastic process is considered where, in each epoch, first, a randomly chosen agent updates their action trying to maximize a proposed utility function, and then agents who have viral exposures beyond their immunity get infected. Our main results discuss asymptotic limiting distributions of both the cardinality of the subset of infected agents and the action profile, considered under various values of two parameters (initial action and immunity profile). We also show that the theoretical distributions are almost always achieved in the first few epochs.

The study of many population growth models is complicated by only partial observation of the underlying stochastic process driving the model. For example, in an epidemic outbreak we might know when individuals show symptoms to a disease and are removed, but not when individuals are infected. Motivated by the above example and the long-established approximation of epidemic processes by branching processes, we explore the number of individuals alive in a time-inhomogeneous branching process with a general phase-type lifetime distribution given only (partial) information on the times of deaths of individuals. Deaths are detected independently with a detection probability that can vary with time and type. We show that the number of individuals alive immediately after the kth detected death can be expressed as the mixture of random variables each of which consists of the sum of k independent zero-modified geometric distributions. Furthermore, in the case of an Erlang lifetime distribution, we derive an easy-to-compute mixture of negative binomial distributions as an approximation of the number of individuals alive immediately after the kth detected death.

This paper analyzes single-item continuous-review inventory models with random supplies in which the inventory dynamic between orders is described by a diffusion process, and a long-term average cost criterion is used to evaluate decisions. The models in this class have general drift and diffusion coefficients and boundary points that are consistent with the notion that demand should tend to reduce the inventory level. Random yield is described by a (probability) transition function which depends on the inventory on hand and the nominal amount ordered; it is assumed to be a distribution with support in the interval determined by the order-from and the nominal order-to locations of the stock level. Using weak convergence arguments involving average expected occupation and ordering measures, conditions are given for the optimality of an (s, S) ordering policy in the general class of policies with finite expected cost. The characterization of the cost of an (s, S) policy as a function of two variables naturally leads to a nonlinear optimization problem over the stock levels s and S, and the existence of an optimizing pair $(s^*,S^*)$ is established under weak conditions. Thus, optimal policies of inventory models with random supplies can (easily) be numerically computed. The range of applicability of the optimality result is illustrated on several inventory models with random yields.

Let $\mathbb{P}_\kappa(n)$ be the probability that n points $z_1,\ldots,z_n$ picked uniformly and independently in $\mathfrak{C}_\kappa$, a regular $\kappa$-gon with area 1, are in convex position, that is, form the vertex set of a convex polygon. In this paper, we compute $\mathbb{P}_\kappa(n)$ up to asymptotic equivalence, as $n\to+\infty$, for all $\kappa\geq 3$, which improves on a famous result of Bárány (Ann. Prob. 27, 1999). The second purpose of this paper is to establish a limit theorem which describes the fluctuations around the limit shape of an n-tuple of points in convex position when $n\to+\infty$. Finally, we give an asymptotically exact algorithm for the random generation of $z_1,\ldots,z_n$, conditioned to be in convex position in $\mathfrak{C}_\kappa$.

This paper obtains logarithmic asymptotics of moderate deviations of the stochastic process of the number of customers in a many-server queue with generally distributed inter-arrival and service times under a heavy-traffic scaling akin to the Halfin–Whitt regime. The deviation function is expressed in terms of the solution to a Fredholm equation of the second kind. A key element of the proof is the large-deviation principle in the scaling of moderate deviations for the sequential empirical process. The techniques of large-deviation convergence and idempotent processes are used extensively.

We provide explicit small-time formulae for the at-the-money implied volatility, skew, and curvature in a large class of models, including rough volatility models and their multi-factor versions. Our general setup encompasses both European options on a stock and VIX options, thereby providing new insights on their joint calibration. The tools used are essentially based on Malliavin calculus for Gaussian processes. We develop a detailed theoretical and numerical analysis of the two-factor rough Bergomi model and provide insights on the interplay between the different parameters for joint SPX–VIX smile calibration.

In Chung–Lu random graphs, a classic model for real-world networks, each vertex is equipped with a weight drawn from a power-law distribution, and two vertices form an edge independently with probability proportional to the product of their weights. Chung–Lu graphs have average distance $O(\log\log n)$ and thus reproduce the small-world phenomenon, a key property of real-world networks. Modern, more realistic variants of this model also equip each vertex with a random position in a specific underlying geometry. The edge probability of two vertices then depends, say, inversely polynomially on their distance.

In this paper we study a generic augmented version of Chung–Lu random graphs. We analyze a model where the edge probability of two vertices can depend arbitrarily on their positions, as long as the marginal probability of forming an edge (for two vertices with fixed weights, one fixed position, and one random position) is as in a Chung–Lu random graph. The resulting class contains Chung–Lu random graphs, hyperbolic random graphs, and geometric inhomogeneous random graphs as special cases.

Our main result is that every random graph model in this general class has the same average distance as a Chung–Lu random graph, up to a factor of $1+o(1)$. This shows in particular that specific choices, such as taking the underlying geometry to be Euclidean, do not significantly influence the average distance. The proof also shows that every random graph model in our class has a giant component and polylogarithmic diameter with high probability.

We solve the non-discounted, finite-horizon optimal stopping problem of a Gauss–Markov bridge by using a time-space transformation approach. The associated optimal stopping boundary is proved to be Lipschitz continuous on any closed interval that excludes the horizon, and it is characterized by the unique solution of an integral equation. A Picard iteration algorithm is discussed and implemented to exemplify the numerical computation and geometry of the optimal stopping boundary for some illustrative cases.

We consider Markov processes that alternate continuous motions and jumps in a general locally compact Polish space. Starting from a mechanistic construction, a first contribution of this article is to provide conditions on the dynamics so that the associated transition kernel forms a Feller semigroup, and to deduce the corresponding infinitesimal generator. As a second contribution, we investigate the ergodic properties in the special case where the jumps consist of births and deaths, a situation observed in several applications including epidemiology, ecology, and microbiology. Based on a coupling argument, we obtain conditions for convergence to a stationary measure with a geometric rate of convergence. Throughout the article, we illustrate our results using general examples of systems of interacting particles in $\mathbb{R}^d$ with births and deaths. We show that in some cases the stationary measure can be made explicit and corresponds to a Gibbs measure on a compact subset of $\mathbb{R}^d$. Our examples include in particular Gibbs measures associated to repulsive Lennard-Jones potentials and to Riesz potentials.

Persistent Betti numbers are a major tool in persistent homology, a subfield of topological data analysis. Many tools in persistent homology rely on the properties of persistent Betti numbers considered as a two-dimensional stochastic process $ (r,s) \mapsto n^{-1/2} (\beta^{r,s}_q ( \mathcal{K}(n^{1/d} \mathcal{X}_n))-\mathbb{E}[\beta^{r,s}_q ( \mathcal{K}( n^{1/d} \mathcal{X}_n))])$. So far, pointwise limit theorems have been established in various settings. In particular, the pointwise asymptotic normality of (persistent) Betti numbers has been established for stationary Poisson processes and binomial processes with constant intensity function in the so-called critical (or thermodynamic) regime; see Yogeshwaran et al. (Prob. Theory Relat. Fields 167, 2017) and Hiraoka et al. (Ann. Appl. Prob. 28, 2018).

In this contribution, we derive a strong stabilization property (in the spirit of Penrose and Yukich, Ann. Appl. Prob. 11, 2001) of persistent Betti numbers, and we generalize the existing results on their asymptotic normality to the multivariate case and to a broader class of underlying Poisson and binomial processes. Most importantly, we show that multivariate asymptotic normality holds for all pairs (r, s), $0\le r\le s<\infty$, and that it is not affected by percolation effects in the underlying random geometric graph.