Advances in Applied Probability最新文献

{"title":"Migration–contagion processes","authors":"F. Baccelli, S. Foss, V. Shneer","doi":"10.1017/apr.2023.17","DOIUrl":"https://doi.org/10.1017/apr.2023.17","url":null,"abstract":"\u0000 Consider the following migration process based on a closed network of N queues with \u0000 \u0000 \u0000 \u0000$K_N$\u0000\u0000 \u0000 customers. Each station is a \u0000 \u0000 \u0000 \u0000$cdot$\u0000\u0000 \u0000 /M/\u0000 \u0000 \u0000 \u0000$infty$\u0000\u0000 \u0000 queue with service (or migration) rate \u0000 \u0000 \u0000 \u0000$mu$\u0000\u0000 \u0000 . Upon departure, a customer is routed independently and uniformly at random to another station. In addition to migration, these customers are subject to a susceptible–infected–susceptible (SIS) dynamics. That is, customers are in one of two states: I for infected, or S for susceptible. Customers can swap their state either from I to S or from S to I only in stations. More precisely, at any station, each susceptible customer becomes infected with the instantaneous rate \u0000 \u0000 \u0000 \u0000$alpha Y$\u0000\u0000 \u0000 if there are Y infected customers in the station, whereas each infected customer recovers and becomes susceptible with rate \u0000 \u0000 \u0000 \u0000$beta$\u0000\u0000 \u0000 . We let N tend to infinity and assume that \u0000 \u0000 \u0000 \u0000$lim_{Nto infty} K_N/N= eta $\u0000\u0000 \u0000 , where \u0000 \u0000 \u0000 \u0000$eta$\u0000\u0000 \u0000 is a positive constant representing the customer density. The main problem of interest concerns the set of parameters of such a system for which there exists a stationary regime where the epidemic survives in the limiting system. The latter limit will be referred to as the thermodynamic limit. We use coupling and stochastic monotonicity arguments to establish key properties of the associated Markov processes, which in turn allow us to give the structure of the phase transition diagram of this thermodynamic limit with respect to \u0000 \u0000 \u0000 \u0000$eta$\u0000\u0000 \u0000 . The analysis of the Kolmogorov equations of this SIS model reduces to that of a wave-type PDE for which we have found no explicit solution. This plain SIS model is one among several companion stochastic processes that exhibit both random migration and contagion. Two of them are discussed in the present paper as they provide variants to the plain SIS model as well as some bounds and approximations. These two variants are the departure-on-change-of-state (DOCS) model and the averaged-infection-rate (AIR) model, which both admit closed-form solutions. The AIR system is a classical mean-field model where the infection mechanism based on the actual population of infected customers is replaced by a mechanism based on some empirical average of the number of infected customers in all stations. The latter admits a product-form solution. DOCS features accelerated migration in that each change of SIS state implies an immediate departure. This model leads to another wave-type PDE that admits a closed-form solution. In this text, the main focus is on the closed stochastic networks and their limits. The open systems consisting of a single station with Poisson input are instrumental in the analysis of the thermodynamic limits and are also of independent interest. This class of SIS dynamics has incarnations in virtually all queueing networks of the literature.","PeriodicalId":53160,"journal":{"name":"Advances in Applied Probability","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2022-08-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47376332","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Interacting nonlinear reinforced stochastic processes: Synchronization or non-synchronization","authors":"I. Crimaldi, P. Louis, I. Minelli","doi":"10.1017/apr.2022.18","DOIUrl":"https://doi.org/10.1017/apr.2022.18","url":null,"abstract":"Abstract 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.","PeriodicalId":53160,"journal":{"name":"Advances in Applied Probability","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2022-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43219538","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"An extended class of univariate and multivariate generalized Pólya processes","authors":"J. Cha","doi":"10.1017/apr.2022.31","DOIUrl":"https://doi.org/10.1017/apr.2022.31","url":null,"abstract":"Abstract 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.","PeriodicalId":53160,"journal":{"name":"Advances in Applied Probability","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2022-07-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48172104","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Branching processes in random environments with thresholds","authors":"Giacomo Francisci, A. Vidyashankar","doi":"10.1017/apr.2023.26","DOIUrl":"https://doi.org/10.1017/apr.2023.26","url":null,"abstract":"\u0000 Motivated by applications to COVID dynamics, we describe a model of a branching process in a random environment \u0000 \u0000 \u0000 \u0000${Z_n}$\u0000\u0000 \u0000 whose characteristics change when crossing upper and lower thresholds. This introduces a cyclical path behavior involving periods of increase and decrease leading to supercritical and subcritical regimes. Even though the process is not Markov, we identify subsequences at random time points \u0000 \u0000 \u0000 \u0000${(tau_j, nu_j)}$\u0000\u0000 \u0000 —specifically the values of the process at crossing times, viz. \u0000 \u0000 \u0000 \u0000${(Z_{tau_j}, Z_{nu_j})}$\u0000\u0000 \u0000 —along which the process retains the Markov structure. Under mild moment and regularity conditions, we establish that the subsequences possess a regenerative structure and prove that the limiting normal distributions of the growth rates of the process in supercritical and subcritical regimes decouple. For this reason, we establish limit theorems concerning the length of supercritical and subcritical regimes and the proportion of time the process spends in these regimes. As a byproduct of our analysis, we explicitly identify the limiting variances in terms of the functionals of the offspring distribution, threshold distribution, and environmental sequences.","PeriodicalId":53160,"journal":{"name":"Advances in Applied Probability","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2022-07-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49365906","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Spectral analysis of bilateral birth–death processes: some new explicit examples","authors":"Manuel D. de la Iglesia","doi":"10.1017/apr.2021.64","DOIUrl":"https://doi.org/10.1017/apr.2021.64","url":null,"abstract":"Abstract 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.","PeriodicalId":53160,"journal":{"name":"Advances in Applied Probability","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2022-06-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46004804","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Gradient estimation for smooth stopping criteria","authors":"B. Heidergott, Yijie Peng","doi":"10.1017/apr.2022.7","DOIUrl":"https://doi.org/10.1017/apr.2022.7","url":null,"abstract":"Abstract 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.","PeriodicalId":53160,"journal":{"name":"Advances in Applied Probability","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2022-06-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45306289","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Central limit theorem for bifurcating markov chains under L2-ergodic conditions","authors":"S. V. Bitseki Penda, Jean-François Delmas","doi":"10.1017/apr.2022.3","DOIUrl":"https://doi.org/10.1017/apr.2022.3","url":null,"abstract":"Abstract 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 \u0000$L^2$\u0000 -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.","PeriodicalId":53160,"journal":{"name":"Advances in Applied Probability","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2022-06-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48714080","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Exponential and gamma form for tail expansions of first-passage distributions in semi-markov processes","authors":"R. Butler","doi":"10.1017/apr.2022.4","DOIUrl":"https://doi.org/10.1017/apr.2022.4","url":null,"abstract":"Abstract 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 \u0000${text{Re}(s)=b}.$\u0000 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.","PeriodicalId":53160,"journal":{"name":"Advances in Applied Probability","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2022-06-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44715614","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Some results on the telegraph process driven by gamma components","authors":"B. Martinucci, Alessandra Meoli, S. Zacks","doi":"10.1017/apr.2021.54","DOIUrl":"https://doi.org/10.1017/apr.2021.54","url":null,"abstract":"Abstract We study the integrated telegraph process \u0000$X_t$\u0000 under the assumption of general distribution for the random times between consecutive reversals of direction. Specifically, \u0000$X_t$\u0000 represents the position, at time t, of a particle moving U time units upwards with velocity c and D time units downwards with velocity \u0000$-c$\u0000 . The latter motions are repeated cyclically, according to independent alternating renewals. Explicit expressions for the probability law of \u0000$X_t$\u0000 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 \u0000$X_t$\u0000 is provided in a closed form. Some expressions for the moment generating function of \u0000$X_t$\u0000 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 \u0000$X_t$\u0000 and disclose its distribution function, specifying the expression for some choices of the distribution of (U, D).","PeriodicalId":53160,"journal":{"name":"Advances in Applied Probability","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2022-06-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44700832","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Optimal consumption with Hindy–Huang–Kreps preferences under nonlinear expectations","authors":"Giorgio Ferrari, Hanwu Li, F. Riedel","doi":"10.1017/apr.2022.5","DOIUrl":"https://doi.org/10.1017/apr.2022.5","url":null,"abstract":"Abstract 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.","PeriodicalId":53160,"journal":{"name":"Advances in Applied Probability","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2022-06-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48932376","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}