This paper investigates the second-order asymptotic tail behavior of stochastic discounted aggregate losses in a discrete-time model, capturing the multidimensional dependence between insurance and financial risks via a multivariate Farlie-Gumbel-Morgenstern distribution. Moving beyond the restrictive assumption of independence or bivariate dependence prevalent in the literature, we derive a second-order asymptotic approximation for the tail probability under the general condition that insurance risks possess a second-order subexponential tail. Our main result provides a closed-form expression that explicitly quantifies the impact of inter-risk dependencies, significantly refining the conventional first-order asymptotics. Simulation studies confirm that the proposed second-order formula markedly enhances estimation accuracy, offering a more reliable tool for capital allocation and risk management.
In this paper we study the potential measures and the Laplace transforms of the occupation times of a refracted-reflected spectrally negative Lévy process when the process is observed at the arrival epochs of two independent Poisson processes. In this case, the rates of observing the underlying process differ in time which deviates from the classical theory of Poissonian observations. Explicit expressions for the so-called Poissonian potential measures and the Poissonian occupation times are derived in terms of (known) scale functions. Other fluctuation identities are also derived.
We extend the basic perishable inventory model with random input stream by adding two realistic new features. In the basic model the items enter the inventory at Poisson times, the demand arrival times are Poisson as well, and each item has a finite deterministic lifespan in the inventory. The new features considered here are (1) batch demands of random size, and (2) finite storage space in the inventory. In Model 1 demands are only accepted if they can be fully satisfied, in Model 2, if a fresh item arrives at a full system it replaces the oldest item present. In both models we analytically determine the steady-state, long-run behavior and the important performance measures in closed form.
This study introduces a nonlinear, multi-factor framework for option pricing by extending the classical Heston-CIR model. Within this framework, both volatility and interest rates are governed by correlated stochastic processes, capturing the full dynamic interdependence among asset returns, volatility, and interest rates in a complex financial environment. To reflect the structural shifts and regime-dependent behaviors observed in real markets, we further incorporate a correlated regime-switching mechanism into these stochastic factors. Despite the increased complexity, the model retains analytical tractability. We derive a generalized moment-generating function under the proposed framework and obtain a closed-form solution for European option prices. Numerical experiments reveal the rich pricing dynamics introduced by the regime-switching structure, while empirical tests demonstrate that the model outperforms the standard Heston-CIR specification, particularly in capturing the nonlinear impacts of correlated economic regimes on option valuation.
In this article, we study the space-time SPDE partial derivative t beta u=-(-Delta)alpha/2u+It1-beta[b(u)+sigma(u)W-center dot], where u=u(t,x) is defined for (t,x)is an element of R+& times;R,beta is an element of(0,1),alpha is an element of(0,2) and W-center dot denotes a space-time white noise. It has long been conjectured that this equation has a unique solution with finite moments under the minimal assumptions of locally Lipschitz coefficients b and sigma with linear growth. We prove that this SPDE is well-posed under the assumptions that the initial condition u0 is bounded and measurable, and the functions b and sigma are locally Lipschitz and have at-most linear growth and some conditions on the Lipschitz constants of the truncated versions of b and sigma. Our results generalize the work of Foondun et al.to a space-time fractional setting.
Software aging, caused by the accumulation of errors and depletion of system resources, degrades the performance of long-running software systems and increases their failure risk. Software rejuvenation alleviates this effect by periodically restoring the system state, and existing rejuvenation strategies are commonly classified as time-based or condition-based. Time-based rejuvenation is easy to implement but lacks adaptability to evolving degradation, whereas condition-based rejuvenation improves efficiency at the expense of frequent inspections and increased operational cost. To overcome these limitations, this paper proposes a hybrid rejuvenation policy that combines scheduled rejuvenation with state-dependent inspections. Under a unified Markovian aging-failure-repair model, we analytically evaluate time-based, condition-based, and hybrid rejuvenation policies using a common stochastic framework. Closed-form expressions for steady-state availability, user-perceived availability, and expected operational cost are derived. Numerical experiments demonstrate that the hybrid policy achieves a more favorable availability-cost tradeoff by retaining the simplicity of time-based rejuvenation while incorporating the adaptive features of condition-based control.
In this paper, under some general conditions, we study the complete l-th moment convergence and complete convergence for maximal randomly weighted sums of arrays of rowwise m-widely orthant dependent (m-WOD, for short) random variables. The results obtained in the paper extend and improve some known results. As an application, we give a result on complete consistency for the weighted estimator in the nonparametric regression model based on m-WOD errors. Additionally, we further perform a numerical simulation to verify the validity of the theoretical results.
Let {Xni;1 <= i <= n,n >= 1} be an array of rowwise random variables in a sub-linear expectation space (Omega,H,E). Suppose that Xnk is independent of (Xn,k+1,& mldr;,Xn,n) for each k=1,& mldr;,n-1,n >= 1. We establish complete convergence and complete moment convergence for maximal weighted sums of arrays of rowwise random variables under optimal moment condition under sub-linear expectations. As an application of our complete convergence theorem, strong laws for weighted sums is obtained. Our results extend and improve the corresponding previous results.
In this paper, we investigate the strong consistency of the least squares (LS) estimators for the unknown parameters beta and theta in a simple linear errors-in-variables (EV) regression model under the assumption of m-widely orthant dependent (m-WOD) random errors. Under weaker conditions, we apply tools such as the strong law of large numbers (SLLN) for randomly weighted sums, truncation techniques, and slicing techniques to derive the sufficient and necessary conditions for the strong consistency. Numerical simulations are conducted to validate our theoretical findings. Additionally, we analyze a real-data example analyzing of diamond prices dataset, and the results demonstrate the practical relevance and applicability of the proposed theory.
Finite mixture models (FMMs) are widely applied in reliability engineering to characterize the lifetime distributions of heterogeneous populations. This paper establishes stochastic comparison results for two FMMs whose component distributions belong to the generalized Marshall-Olkin Topp Leone-G (MOTL-G) family, which uses a scale family as the baseline distribution. The analysis is conducted within the framework of majorization, chain majorization, and unordered majorization orders. We first derive sufficient conditions under which the usual stochastic order and the hazard rate order hold between two FMMs, in terms of majorization and chain majorization orders applied to the shape, scale, and mixing proportion parameters. Subsequently, we develop sufficient conditions based on unordered majorization orders for the usual stochastic order between two FMMs. Using aluminum alloy fatigue life data, we validate the flexibility of the proposed model and successfully identify its underlying heterogeneous structure. The theoretical findings are illustrated and validated through numerical examples and counterexamples.
In this paper, we prove the existence and uniqueness of solutions for two classes of doubly reflected backward stochastic differential equations (DRBSDEs) driven by pure jump Markov and jump semi-Markov processes. The analysis is based on the Snell envelope technique and a penalization method.
This paper presents a sparse representation method for stochastic signals based on the complete dictionary constructed from the Szeg & odblac; kernel. The study extends the theoretical framework proposed by Qian et al. By designing a complete dictionary of the Szeg & odblac; kernel, we achieve an adaptive sparse expansion of stochastic signals. A rigorous convergence theorem is established, and a practical algorithm is developed. The proposed methodology is further applied to numerical simulations of stochastic processes, demonstrating significant advantages in the representation of random signals.
The article explores the asymptotic behavior of the expected number of drawings in the Coupon Collector's Problem with group-drawing under the uniform distribution. In this variant, each draw consists of a package of s distinct coupons selected uniformly at random from a set of n coupons. We focus on three regimes of the package size s: (i) constant s, (ii) s proportional to n, and (iii) s "very close" to n. For each case, we provide precise asymptotic expressions for the expected collection time.
In 1978, Bramson constructed a branching random walk where particles branch with mean m and move one step to the right with probability 1-1/m, or remain at the same site with probability 1/m. It has been proved that M-n, the minimal position of particles at time n, grows extremely slowly, of order log log n. When particles reproduce according to the same offspring distribution but move according to a random walk in random environment (RWRE), i.e., particles at position i move one step to the right with probability 1-omega(i) or stay at the same site with probability omega(i), where (omega(i))(i)is an element of N are i.i.d. random variables and eta denotes the probability distribution of omega:=(omega(i))(i is an element of N). Nakashima (2013) showed that under the conditions ess sup(omega)omega 0=1/m and eta(omega(0)=1/m)=0, and further assumptions on the environment, E[M-n] is of order n(alpha/alpha+1) for some alpha > 0. In this article, we consider the case where ess sup(omega)omega(0)=1/m but eta(omega(0)=1/m)>0. We prove that, for almost all environment omega, M-n/log log n -> 1/eta(omega(0)=1/m)log 2' P-omega-a.e. Thus, the growth rate of M-n is again of order log log n, as in Bramson (1978), while the effect of the random environment is captured by the parameter eta(omega(0)=1/m). To clarify the effect of the random environment, it is necessary to characterize the specific order of Z(j), the number of particles that ever jump from position j-1 to j. We prove that (Z(j))(j is an element of N) is a heavy-tailed branching process in random environment satisfying E[log(E(omega)Z1)] = infinity, in contrast to the case eta(omega(0)=1/m)=0 studied by Nakashima (2013), where E[log(E(omega)Z1)] < infinity. Since the classical Kesten-Stigum Theorem and Seneta-Heyde Theorem do not apply in this setting, the study of the limit behavior of (Z(j))(j is an element of N) is of independent interest.
In this article, we provide the sufficient and necessary conditions for the strong consistency of least squares estimators in simple linear errors-in-variables regression models with the m-END random errors. The main methodologies employed are the Kolmogorov-type strong law of large numbers and Rosenthal-type moment inequality for m-END random variables. The results extend the corresponding ones in the literature. A numerical simulation is also performed to support the theoretical results based on the finite samples.
Let W-1,..., Wn be non-negative random variables. We consider an undirected random graph model on the node set {1,. ..,n}, where two nodes i < j are adjacent if W-i < W-j. In our setting, the Wi's are independent but not necessarily identically distributed, resulting in a model that generalizes the classical random permutation graphs. The model exhibits a certain dependence among the edges. Moreover, when nodes have physical interpretations- such as points on the real line R with node i located at position x = i-the model gains spatial structure and becomes, in particular, distance-dependent. We derive theoretical results on degree distributions, the number of isolated vertices, and the number of close neighbors. Simulation-based observations are also provided for the average clustering and the global efficiency.
We give functional laws of large numbers for a class of marked Hawkes processes and marked compound Hawkes processes with a general mark space. Our results provide some complement to those presented in e.g., Bacry et al. [1 c1] and Horst and Xu [6 c6]. As an example, we provide an application to analysis of time limit of an insurance ruin process.
We address a conjecture of Schilling concerning the optimality of the uniform distribution in the generalized Coupon Collector's Problem (CCP) where, in each round, a subset (package) of s coupons is drawn from a total of n distinct coupons. While the classical CCP (with single-coupon draws) is well understood, the group-draw variant, where packages of size s are drawn, presents new challenges and has applications in areas such as biological network models. Consider the set of all distributions over the collection of (ns) packages of size s. Schilling showed that, for s=n-1, the uniform distribution yields the minimal expected time for collecting all coupons. She further conjectured that, for 2 <= s <= n-2, the uniform distribution does not yield the minimum. We prove Schilling's conjecture in full by presenting "natural" non-uniform distributions yielding strictly lower expected collection times. Explicit formulas are provided for the expected number of rounds under these and related distributions.
In this paper, we introduce a novel model for random hypergraphs based on weighted random connection models. In accordance with the standard theory for hypergraphs, this model is constructed from a bipartite graph. In our stochastic model, both vertex sets of this bipartite graph form marked Poisson point processes, and the connection radius is inversely proportional to a product of suitable powers of the marks. Hence, our model is a common generalization of weighted random connection models and AB random geometric graphs. For this hypergraph model, we investigate the limit theory of various graph-theoretic and topological characteristics, including higher-order degree distributions, Betti numbers of the associated Dowker complex, and simplex counts. In particular, for the latter quantity, we identify regimes of convergence to normal and to stable distribution depending on the heavy-tailedness of the weight distribution. We conclude our investigation with a simulation study and an application to the collaboration network extracted from the arXiv dataset.
