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This paper examines a model involving two dynamic Erd\H{o}s-R\'enyi random graphs that {evolve} in parallel, where the edges in each of these graphs alternate between being present and absent according to specific on- and off-time distributions. In our setup there is {\it regime switching}, in that the graph that we observe depends on the mode of the {\it background process} at that point in time, this background process being modeled as an alternating renewal process. In our setup we only have access to aggregate quantities such as the number of active edges or the counts of specific subgraphs (such as stars or {complete graphs}) in the observed graph; importantly, we do not observe the mode. The objective is to estimate the on- and off-time distributions of the edges in each of the two dynamic Erd\H{o}s-R\'enyi random graphs, as well as the distribution of time spent in each of the two modes. By employing parametric models for the on- and off-times and the background process, we develop a method of moments approach to estimate the relevant parameters. Numerical experiments are conducted to demonstrate the effectiveness of the proposed method in recovering these parameters.

We consider two-opinion voter models on dense dynamic random graphs. Our goal is to understand and describe the occurrence of consensus versus polarisation over long periods of time. The former means that all vertices have the same opinion, the latter means that the vertices split into two communities with different opinions and few disagreeing edges. We consider three models for the joint dynamics of opinions and graphs: one with a one-way feedback and two which are co-evolutionary, i.e., with a two-way feedback. In the first model only coexistence is attainable, meaning that both opinions survive, but with the presence of many disagreeing edges. In the second model only consensus prevails, while in the third model polarisation is possible. Our main results are functional laws of large numbers for the densities of the two opinions, functional laws of large numbers for the dynamic random graphs in the space of graphons, and a characterisation of the limiting densities in terms of Beta-distributions. Our results are supported by simulations. To prove our results we develop a novel method that involves coupling the co-evolutionary process to a mimicking process with one-way feedback. We expect that this method can be extended to other dense co-evolutionary models., Comment: 39 pages

This paper concerns discrete-time occupancy processes on a finite graph. Our results can be formulated in two theorems, which are stated for vertex processes, but also applied to edge process (e.g., dynamic random graphs). The first theorem shows that concentration of local state averages is controlled by a random walk on the graph. The second theorem concerns concentration of polynomials of the vertex states. For dynamic random graphs, this allows to estimate deviations of edge density, triangle density, and more general subgraph densities. Our results only require Lipschitz continuity and hold for both dense and sparse graphs., Comment: 14 pages

In this paper we consider an M/G/1-type queue fed by a finite customer-pool. In terms of transforms, we characterize the time-dependent distribution of the number of customers and the workload, as well as the associated waiting times.

We introduce a spatiotemporal self-exciting point process $(N_t(x))$, boundedly finite both over time $[0,\infty)$ and space $\mathscr X$, with excitation structure determined by a graphon $W$ on $\mathscr{X}^2$. This graphon Hawkes process generalizes both the multivariate Hawkes process and the Hawkes process on a countable network, and despite being infinite-dimensional, it is surprisingly tractable. After proving existence, uniqueness and stability results, we show, both in the annealed and in the quenched case, that for compact, Euclidean $\mathscr X\subset\mathbb R^m$, any graphon Hawkes process can be obtained as the suitable limit of $d$-dimensional Hawkes processes $\tilde N^d$, as $d\to\infty$. Furthermore, in the stable regime, we establish an FLLN and an FCLT for our infinite-dimensional process on compact $\mathscr X\subset\mathbb R^m$, while in the unstable regime we prove divergence of $N_T(\mathscr X)/T$, as $T\to\infty$. Finally, we exploit a cluster representation to derive fixed-point equations for the Laplace functional of $N$, for which we set up a recursive approximation procedure. We apply these results to show that, starting with multivariate Hawkes processes $\tilde N^d_t$ converging to stable graphon Hawkes processes, the limits $d\to\infty$ and $t\to\infty$ commute.

This paper studies a novel stochastic compartmental model that describes the dynamics of trust in society. The population is split into three compartments representing levels of trust in society: trusters, skeptics and doubters. The focus lies on assessing the long-term dynamics, under `bounded confidence' i.e., trusters and doubters do not communicate). We state and classify the stationary points of the system's mean behavior. We find that an increase in life-expectancy, and a greater population may increase the proportion of individuals who lose their trust completely. In addition, the relationship between the rate at which doubters convince skeptics to join their cause and the expected number of doubters is not monotonic -- it does not always help to be more convincing to ensure the survival of your group. We numerically illustrate the workings of our analysis. Because the study of stochastic compartmental models for social dynamics is not common, we in particular shed light on the limitations of deterministic compartmental models. In our experiments we make use of fluid and diffusion approximation techniques as well as Gillespie simulation., Comment: 22 pages including references and appendices, 9 Figures, comments welcome

In this paper we consider a dynamic version of the Erd\H{o}s-R\'{e}nyi random graph, in which edges independently appear and disappear in time, with the on- and off times being exponentially distributed. The focus lies on the evolution of the principle eigenvalue of the adjacency matrix in the regime that the number of vertices grows large. The main result is a functional central limit theorem, which displays that the principal eigenvalue essentially inherits the characteristics of the dynamics of the individual edges.

In this paper we analyze the distribution of the Age of Information (AoI) of a tagged data stream sharing a processor with a set of other data streams. We do so in the highly general setting in which the interarrival times pertaining to the tagged stream can have any distribution, and also the service times of both the tagged stream and the background stream are generally distributed. The packet arrival times of the background process are assumed to constitute a Poisson process, which is justified by the fact that it typically is a superposition of many relatively homogeneous streams. The first major contribution is that we derive an expression for the Laplace-Stieltjes transform of the AoI in the resulting GI+M/GI+GI/1 model. Second, we use stochastic ordering techniques to identify tight stochastic bounds on the AoI. In addition, when approximating the tagged stream's inter-generation times through a phase-type distribution (which can be done at any precision), we present a computational algorithm for the mean AoI. As illustrated through a sequence of numerical experiments, the analysis enables us to assess the impact of background traffic on the AoI of the tagged stream.

We study the interpersonal trust of a population of agents, asking whether chance may decide if a population ends up in a high trust or low trust state. We model this by a discrete time, random matching stochastic coordination game. Agents are endowed with an exponential smoothing learning rule about the behaviour of their neighbours. We find that, with probability one in the long run the whole population either always cooperates or always defects. By simulation we study the impact of the distributions of the payoffs in the game and of the exponential smoothing learning (memory of the agents). We find, that as the agent memory increases or as the size of the population increases, the actual dynamics start to resemble the expectation of the process. We conclude that it is indeed possible that different populations may converge upon high or low trust between its citizens simply by chance, though the game parameters (context of the society) may be quite telling., Comment: 17 pages, 4 figures

In this paper we consider a dynamic Erd\H{o}s-R\'enyi graph in which edges, according to an alternating renewal process, change from present to absent and vice versa. The objective is to estimate the on- and off-time distributions while only observing the aggregate number of edges. This inverse problem is dealt with, in a parametric context, by setting up an estimator based on the method of moments. We provide conditions under which the estimator is asymptotically normal, and we point out how the corresponding covariance matrix can be identified. It is also demonstrated how to adapt the estimation procedure if alternative subgraph counts are observed, such as the number of wedges or triangles.

Setting up optimal appointment schedules requires the computation of an inherently involved objective function, typically requiring distributional knowledge of the clients' waiting times and the server's idle times (as a function of the appointment times of the individual clients). A frequently used idea is to approximate the clients' service times by their phase-type counterpart, thus leading to explicit expressions for the waiting-time and idle-time distributions. This method, however, requires the evaluation of the matrix exponential of potentially large matrices, which already becomes prohibitively slow from, say, 20 clients on. In this paper we remedy this issue by recursively approximating the distributions involved relying on a two-moments fit. More specifically, we approximate the sojourn time of each of the clients by a low-dimensional phase-type, Weibull or Lognormal random variable with the desired mean and variance. Our computational experiments show that this elementary, yet highly accurate, technique facilitates the evaluation of optimal appointment schedules even if the number of clients is large. The three ways to approximate the sojourn-time distribution turn out to be roughly equally accurate, except in certain specific regimes, where the low-dimensional phase-type fit performs well across all instances considered. As this low-dimensional phase-type fit is by far the fastest of the three alternatives, it is the approximation that we recommend., Comment: 21 pages, 8 figures

This paper aims to address the integrated routing and appointment scheduling (RAS) problem for a single service provider. The RAS problem is an operational challenge faced by operators that provide services requiring home attendance, such as grocery delivery, home healthcare, or maintenance services. While considering the inherently random nature of service and travel times, the goal is to minimize a weighted sum of the operator's travel times and idle time, and the client's waiting times. To handle the complex search space of routing and appointment scheduling decisions, we propose a queueing-based approach to effectively deal with the appointment scheduling decisions. We use two well-known approximations from queueing theory: first, we use an approach based on phase-type distributions to accurately approximate the objective function, and second, we use an heavy-traffic approximation to derive an efficient procedure to obtain good appointment schedules. Combining these two approaches results in a fast and sufficiently accurate hybrid approximation, thus essentially reducing RAS to a routing problem. Moreover, we propose the use a simple yet effective large neighborhood search metaheuristic to explore the space of routing decisions. The effectiveness of our proposed methodology is tested on benchmark instances with up to 40 clients, demonstrating an efficient and accurate methodology for integrated routing and appointment scheduling., Comment: 25 pages, 10 figures

We study a multi-server queueing system with a periodic arrival rate and customers whose joining decision is based on their patience and a delay proxy. Specifically, each customer has a patience level sampled from a common distribution. Upon arrival, they receive an estimate of their delay before joining service and then join the system only if this delay is not more than their patience, otherwise they balk. The main objective is to estimate the parameters pertaining to the arrival rate and patience distribution. Here the complication factor is that this inference should be performed based on the observed process only, i.e., balking customers remain unobserved. We set up a likelihood function of the state dependent effective arrival process (i.e., corresponding to the customers who join), establish strong consistency of the MLE, and derive the asymptotic distribution of the estimation error. Due to the intrinsic non-stationarity of the Poisson arrival process, the proof techniques used in previous work become inapplicable. The novelty of the proving mechanism in this paper lies in the procedure of constructing i.i.d. objects from dependent samples by decomposing the sample path into i.i.d. regeneration cycles. The feasibility of the MLE-approach is discussed via a sequence of numerical experiments, for multiple choices of functions which provide delay estimates. In particular, it is observed that the arrival rate is best estimated at high service capacities, and the patience distribution is best estimated at lower service capacities., Comment: 39 pages, 34 figures

We study the stochastic properties of the area under some function of the difference between (i) a spectrally positive L\'evy process $W_t^x$ that jumps to a level $x>0$ whenever it hits zero, and (ii) its reflected version $W_t$. Remarkably, even though the analysis of each of these areas is challenging, we succeed in attaining explicit expressions for their difference. The main result concerns the Laplace-Stieltjes transform of the integral $A_x$ of (a function of) the distance between $W_t^x$ and $W_t$ until $W_t^x$ hits zero. This result is extended in a number of directions, including the area between $A_x$ and $A_y$ and a Gaussian limit theorem. We conclude the paper with an inventory problem for which our results are particularly useful.

Let $J(\cdot)$ be a compound Poisson process with rate $\lambda>0$ and a jumps distribution $G(\cdot)$ concentrated on $(0,\infty)$. In addition, let $V$ be a random variable which is distributed according to $G(\cdot)$ and independent from $J(\cdot)$. Define a new process $W(t)\equiv W_V(t)\equiv V+J(t)-t$, $t\geq0$ and let $\tau_V$ be the first time that $W(\cdot)$ hits the origin. A long-standing open problem due to Iglehart (1971) and Cohen (1979) is to derive the moments of the functional $\int_0^\tau W(t)\,{\rm d}t$ in terms of the moments of $G(\cdot)$ and $\lambda$. In the current work, we solve this problem in much greater generality, i.e., first by letting $J(\cdot)$ belong to a wide class of spectrally-positive L\'evy processes and secondly, by considering more general class of functionals. We also supply several applications of the existing results, e.g., in studying the process $x\mapsto \int_0^{\tau_x}W_x(t)\,{\rm d}t$ defined on $x\in[0,\infty)$.

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, Cram\'er-Lundberg type asymptotics are established. Also, an importance sampling based Monte Carlo algorithm is proposed, and is shown to be logarithmically efficient.

We introduce a variant of the Hawkes-fed birth-death process, in which the conditional intensity does not increase at arrivals, but at departures from the system. Since arrivals cause excitation after a delay equal to their lifetimes, we call this a delayed Hawkes process. We introduce a general family of models admitting a cluster representation containing the Hawkes, delayed Hawkes and ephemerally self-exciting processes as special cases. For this family of models, as well as their nonlinear extensions, we prove existence, uniqueness and stability. Our family of models satisfies the same FCLT as the classical Hawkes process; however, we describe a scaling limit for the delayed Hawkes process in which sojourn times are stretched out by a factor $\sqrt T$, after which time gets contracted by a factor $T$. This scaling limit highlights the effect of sojourn-time dependence. The cluster representation renders our family of models tractable, allowing for transform characterisation by a fixed-point equation and for an analysis of heavy-tailed asymptotics. In the Markovian case, for a multivariate network of delayed Hawkes birth-death processes, an explicit recursive procedure is presented to calculate the $d$th-order moments analytically. Finally, we compare the delayed Hawkes process to the regular Hawkes process in the stochastic ordering, which enables us to describe stationary distributions and heavy-traffic behaviour., Comment: 38 pages, 1 figure

We study the problem of an agent continuously faced with the decision of placing or not placing trust in an institution. The agent makes use of Bayesian learning in order to estimate the institution's true trustworthiness and makes the decision to place trust based on myopic rationality. Using elements from random walk theory, we explicitly derive the probability that such an agent ceases placing trust at some point in the relationship, as well as the expected time spent placing trust conditioned on their discontinuation thereof. We then continue by modelling two truster agents, each in their own relationship to the institution. We consider two natural models of communication between them. In the first (``observable rewards'') agents disclose their experiences with the institution with one another, while in the second (``observable actions'') agents merely witness the actions of their neighbour, i.e., placing or not placing trust. Under the same assumptions as in the single agent case, we describe the evolution of the beliefs of agents under these two different communication models. Both the probability of ceasing to place trust and the expected time in the system elude explicit expressions, despite there being only two agents. We therefore conduct a simulation study in order to compare the effect of the different kinds of communication on the trust dynamics. We find that a pair of agents in both communication models has a greater chance of learning the true trustworthiness of an institution than a single agent. Communication between agents promotes the formation of long term trust with a trustworthy institution as well as the timely exit from a trust relationship with an untrustworthy institution. Contrary to what one might expect, we find that having less information (observing each other's actions instead of experiences) can sometimes be beneficial to the agents.

Consider a L\'evy process $Y(t)$ over an exponentially distributed time $T_\beta$ with mean $1/\beta$. We study the joint distribution of the running maximum $\bar{Y}(T_\beta)$ and the time epoch $G(T_\beta$) at which this maximum last occurs. Our main result is a fluctuation-theoretic distributional equality: the vector ($\bar{Y}(T_\beta),G(T_\beta)$) can be written as a sum of two independent vectors, the first one being ($\bar{Y}(T_{\beta+\omega}),G(T_{\beta+\omega})$) and the second one being the running maximum and corresponding time epoch under the restriction that the L\'evy process is only observed at Poisson($\omega$) inspection epochs (until $T_\beta$). We first provide an analytic proof for this remarkable decomposition, and then a more elementary proof that gives insight into the occurrence of the decomposition and into the fact that $\omega$ only appears in the right hand side of the decomposition. The proof technique underlying the more elementary derivation also leads to further generalizations of the decomposition, and to some fundamental insights into a generalization of the well known Lindley recursion.

In this paper, we establish a large deviations principle for a multivariate compound process induced by a multivariate Hawkes process with random marks. Our proof hinges on showing essential smoothness of the limiting cumulant of the multivariate compound process, resolving the inherent complication that this cumulant is implicitly characterized through a fixed-point representation. We employ the large deviations principle to derive logarithmic asymptotic results on the marginal ruin probabilities of the associated multivariate risk process. We also show how to conduct rare event simulation in this multivariate setting using importance sampling and prove the asymptotic efficiency of our importance sampling based estimator. The paper is concluded with a systematic assessment of the performance of our rare event simulation procedure.

In route selection problems, the driver's personal preferences will determine whether she prefers a route with a travel time that has a relatively low mean and high variance over one that has relatively high mean and low variance. In practice, however, such risk aversion issues are often ignored, in that a route is selected based on a single-criterion Dijkstra-type algorithm. In addition, the routing decision typically does not take into account the uncertainty in the estimates of the travel time's mean and variance. This paper aims at resolving both issues by setting up a framework for travel time estimation. In our framework, the underlying road network is represented as a graph. Each edge is subdivided into multiple smaller pieces, so as to naturally model the statistical similarity between road pieces that are spatially nearby. Relying on a Bayesian approach, we construct an estimator for the joint per-edge travel time distribution, thus also providing us with an uncertainty quantification of our estimates. Our machinery relies on establishing limit theorems, making the resulting estimation procedure robust in the sense that it effectively does not assume any distributional properties. We present an extensive set of numerical experiments that demonstrate the validity of the estimation procedure and the use of the distributional estimates in the context of data-driven route selection.

We consider a class of graph-valued stochastic processes in which each vertex has a type that fluctuates randomly over time. Collectively, the paths of the vertex types up to a given time determine the probabilities that the edges are active or inactive at that time. Our focus is on the evolution of the associated empirical graphon in the limit as the number of vertices tends to infinity, in the setting where fluctuations in the graph-valued process are more likely to be caused by fluctuations in the vertex types than by fluctuations in the states of the edges given these types. We derive both sample-path large deviation principles and convergence of stochastic processes. We demonstrate the flexibility of our approach by treating a class of stochastic processes where the edge probabilities depend not only on the fluctuations in the vertex types but also on the state of the graph itself., Comment: 41 pages, 4 figures

Due to recurrent (e.g. daily or weekly) patterns and non-recurrent disruptions (e.g. caused by incidents), travel times in road networks are time-dependent and inherently random. This is challenging for travelers planning a future trip, aiming to ensure on-time arrival at the destination, while also trying to limit the total travel-time budget spent. The focus of this paper lies on determining their optimal departure time: the latest time of departure for which a chosen on-time arrival probability can be guaranteed. To model the uncertainties in the network, a Markovian background process is used, tracking events affecting the driveable vehicle speeds on the links, thus enabling us to incorporate both recurrent and non-recurrent effects. It allows the evaluation of the travel-time distribution, given the state of this process at departure, on each single link. Then, a computationally efficient algorithm is devised that uses these individual link travel-time distributions to obtain the optimal departure time for a given path or origin-destination pair. Since the conditions in the road network, and thus the state of the background process, may change between the time of request and the advised time of departure, we consider an online version of this procedure as well, in which the traveler receives departure time updates while still at the origin. Finally, numerical experiments are conducted to exemplify a selection of properties of the optimal departure time and, moreover, quantify the performance of the presented algorithms in an existing road network -- the Dutch highway network.

To improve the routing decisions of individual drivers and the management policies designed by traffic operators, one needs reliable estimates of travel time distributions. Since congestion caused by both recurrent patterns (e.g., rush hours) and non-recurrent events (e.g., traffic incidents) leads to potentially substantial delays in highway travel times, we focus on a framework capable of incorporating both effects. To this end, we propose to work with the Markovian Velocity model (MVM), based on an environmental background process that tracks both random and (semi-)predictable events affecting the vehicle speeds in a highway network. We show how to operationalize this flexible data-driven model in order to obtain the travel time distribution for a vehicle departing at a known day and time to traverse a given path. Specifically, we detail how to structure the background process and set the speed levels corresponding to the different states of this process. First, for the inclusion of non-recurrent events, we study incident data to describe the random durations of the incident and inter-incident times. Both of them depend on the time of day, but we identify periods in which they can be considered time-independent. Second, for an estimation of the speed patterns in both incident and inter-incident regime, loop detector data for each of the identified periods is studied. In numerical examples that use road network detector data of the Dutch highway network, we obtain the travel time distribution estimates that arise under different traffic regimes, and illustrate the advantages compared to traditional travel-time prediction methods.

A Chung-Lu random graph is an inhomogeneous Erd\H{o}s-R\'enyi random graph in which vertices are assigned average degrees, and pairs of vertices are connected by an edge with a probability that is proportional to the product of their average degrees, independently for different edges. We derive a central limit theorem for the principal eigenvalue and the components of the principal eigenvector of the adjacency matrix of a Chung-Lu random graph. Our derivation requires certain assumptions on the average degrees that guarantee connectivity, sparsity and bounded inhomogeneity of the graph.

Externalities are the costs that a user of a common resource imposes on others. For example, consider a FCFS M/G/1 queue and a customer with service demand of $x\geq0$ minutes who arrived into the system when the workload level was $v\geq0$ minutes. Let $E_v(x)$ be the total waiting time which could be saved if this customer gave up on his service demand. In this work, we analyse the \textit{externalities process} $E_v(\cdot)=\left\{E_v(x):x\geq0\right\}$. It is shown that this process can be represented by an integral of a (shifted in time by $v$ minutes) compound Poisson process with positive discrete jump distribution, so that $E_v(\cdot)$ is convex. Furthermore, we compute the LST of the finite-dimensional distributions of $E_v(\cdot)$ as well as its mean and auto-covariance functions. We also identify conditions under which, a sequence of normalized externalities processes admits a weak convergence on $\mathcal{D}[0,\infty)$ equipped with the uniform metric to an integral of a (shifted in time by $v$ minutes) standard Wiener process. Finally, we also consider the extended framework when $v$ is a general nonnegative random variable which is independent from the arrival process and the service demands. This leads to a generalization of an existing result from a previous work of Haviv and Ritov (1998).

In this paper we establish a necessary and sufficient stability condition for a stochastic ring network. Such networks naturally appear in a variety of applications within communication, computer, and road traffic systems. They typically involve multiple customer types and some form of priority structure to decide which customer receives service. These two system features tend to complicate the issue of identifying a stability condition, but we demonstrate how the ring topology can be leveraged to solve the problem., Comment: 25 pages, 2 figures; v2: revamped Section 3.1, rewritten Section 6.1, expanded Section 7, added two figures and two references, streamlined the proof of Theorem 5.6, additional minor edits throughout

Our goal is to estimate the characteristic exponent of the input to a L\'evy-driven storage system from a sample of equispaced workload observations. The estimator relies on an approximate moment equation associated with the Laplace-Stieltjes transform of the workload at exponentially distributed sampling times. The estimator is pointwise consistent for any observation grid. Moreover, a high frequency sampling scheme yields asymptotically normal estimation errors for a class of input processes. A resampling scheme that uses the available information in a more efficient manner is suggested and assessed via simulation experiments.

Despite measures to reduce congestion, occurrences of both recurrent and non-recurrent congestion cause large delays in road networks with important economic implications. Educated use of Intelligent Transportation Systems (ITS) can significantly reduce travel times. We focus on a dynamic stochastic shortest path problem: our objective is to minimize the expected travel time of a vehicle, assuming the vehicle may adapt the chosen route while driving. We introduce a new stochastic process that incorporates ITS information to model the uncertainties affecting congestion in road networks. A Markov-modulated background process tracks traffic events that affect the speed of travelers. The resulting continuous-time routing model allows for correlation between velocities on the arcs and incorporates both recurrent and non-recurrent congestion. Obtaining the optimal routing policy in the resulting semi-Markov decision process using dynamic programming is computationally intractable for realistic network sizes. To overcome this, we present the EDSGER* algorithm, a Dijkstra-like shortest path algorithm that can be used dynamically with real-time response. We develop additional speed-up techniques that reduce the size of the network model. We quantify the performance of the algorithms by providing numerical examples that use road network detector data for The Netherlands.

This paper considers the Cram\'er-Lundberg model, with the additional feature that the number of clients can fluctuate over time. Clients arrive according to a Poisson process, where the times they spend in the system form a sequence of independent and identically distributed non-negative random variables. While in the system, every client generates claims and pays premiums. In order to describe the model's rare-event behaviour, we establish a sample-path large-deviation principle. This describes the joint rare-event behaviour of the reserve-level process and the client-population size process. The large-deviation principle can be used to determine the decay rate of the time-dependent ruin probability as well as the most likely path to ruin. Our results allow us to determine whether the chance of ruin is greater with more or with fewer clients and, more generally, to determine to what extent a large deviation in the reserve-level process can be attributed to an unusual outcome of the client-population size process., Comment: 28 pages, 2 figures; v2-minor revision

We consider a L\'evy process $Y(t)$ that is not permanently observed, but rather inspected at Poisson($\omega$) moments only, over an exponentially distributed time $T_\beta$ with parameter $\beta$. The focus lies on the analysis of the distribution of the running maximum at such inspection moments up to $T_\beta$, denoted by $Y_{\beta,\omega}$. Our main result is a decomposition: we derive a remarkable distributional equality that contains $Y_{\beta,\omega}$ as well as the running maximum process $\bar Y(t)$ at the exponentially distributed times $T_\beta$ and $T_{\beta+\omega}$. Concretely, $\overline{Y}(T_\beta)$ can be written the sum of the two independent random variables that are distributed as $Y_{\beta,\omega}$ and $\overline{Y}(T_{\beta+\omega})$. The distribution of $Y_{\beta,\omega}$ can be identified more explicitly in the two special cases of a spectrally positive and a spectrally negative L\'evy process. As an illustrative example of the potential of our results, we show how to determine the asymptotic behavior of the bankruptcy probability in the Cram\'er-Lundberg insurance risk model.

This paper considers population processes in which general, not necessarily Markovian, multivariate Hawkes processes dictate the stochastic arrivals. We establish results to determine the corresponding time-dependent joint probability distribution, allowing for general intensity decay functions, general intensity jumps, and general sojourn times. We obtain an exact, full characterization of the time-dependent joint transform of the multivariate population process and its underlying intensity process in terms of a fixed-point representation and corresponding convergence results. We also derive the asymptotic tail behavior of the population process and its underlying intensity process in the setting of heavy-tailed intensity jumps. By exploiting the results we establish, arbitrary joint spatial-temporal moments and other distributional properties can now be readily evaluated using standard transform differentiation and inversion techniques, and we illustrate this in a few examples.

This paper addresses allocation methodologies for a risk measure inherited from ruin theory. Specifically, we consider a dynamic value-at-risk (VaR) measure defined as the smallest initial capital needed to ensure that the ultimate ruin probability is less than a given threshold. We introduce an intuitively appealing, novel allocation method, with a focus on its application to capital reserves which are determined through the dynamic value-at-risk (VaR) measure. Various desirable properties of the presented approach are derived including a limit result when considering a large time horizon and the comparison with the frequently used gradient allocation method. In passing, we introduce a second allocation method and discuss its relation to the other allocation approaches. A number of examples illustrate the applicability and performance of the allocation approaches.

Formal models of opinion formation commonly represent an individual's opinion by a value on a fixed opinion interval. We propose an alternative modeling method wherein interpretation is only provided to the relative positions of opinions vis-\`a-vis each other. This method is then considered in a similar setting as the discrete-time Altafini model (an extension of the well-known DeGroot model), but with more general influence weights. Even in a linear framework, the model can describe, in the long run, polarization, dynamics with a periodic pattern, and (modulus) consensus formation. In addition, in our alternative approach key characteristics of the opinion dynamic can be derived from real-valued square matrices of influence weights, which immediately allows one to transfer matrix theory insights to the field of opinion formation dynamics under more relaxed conditions than in the DeGroot or discrete-time Altafini models. A few specific themes are covered: (i) We demonstrate how stable patterns in relative opinion dynamics are identified which are hidden when opinions are considered in an absolute opinion framework. (ii) For the two-agent case, we provide an exhaustive closed-form description of the relative opinion model's dynamic in the long run. (iii) We explore group dynamics analytically, in particular providing a non-trivial condition under which a subgroup's asymptotic behavior carries over to the entire population., Comment: 40 pages

This paper considers appointment scheduling in a setting in which at every client arrival the schedule of all future clients can be adapted. Starting our analysis with an explicit treatment of the case of exponentially distributed service times, we then develop a phase-type-based approach to also cover cases in which the service times' squared coefficient of variation differs from 1. The approach relies on dynamic programming, with the state information being the number of clients waiting, the elapsed service time of the client in service, and the number of clients still to be scheduled. The use of dynamic schedules is illustrated through a set of numerical experiments, showing (i) the effect of wrongly assuming exponentially distributed service times, and (ii) the gains (over static schedules, that is) achieved by rescheduling.

We consider a dynamic Erd\H{o}s-R\'enyi random graph (ERRG) on $n$ vertices in which each edge switches on at rate $\lambda$ and switches off at rate $\mu$, independently of other edges. The focus is on the analysis of the evolution of the associated empirical graphon in the limit as $n\to\infty$. Our main result is a large deviation principle (LDP) for the sample path of the empirical graphon observed until a fixed time horizon. The rate is $\binom{n}{2}$, the rate function is a specific action integral on the space of graphon trajectories. We apply the LDP to identify (i) the most likely path that starting from a constant graphon creates a graphon with an atypically large density of $d$-regular subgraphs, and (ii) the mostly likely path between two given graphons. It turns out that bifurcations may occur in the solutions of associated variational problems.