. 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.

. 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.

. 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−ω i or stay at the same site with probability ω i , where ( ω i ) i ∈ N are i . i . d . random variables and η denotes the probability distribution of ω := ( ω i ) i ∈ N . Nakashima (2013) showed that under the conditions ess sup ω ω 0 = 1 m and η ( ω 0 = 1 m ) = 0 , and further assumptions on the environment, 𝔼[M n ] is of order n α α + 1 for some α > 0. In this article, we consider the case where ess sup ω ω 0 = 1 m but η ( ω 0 = 1 m ) > 0 . We prove that, for almost all environment ω, M n log ⁡ log ⁡ n → 1 η ( ω 0 = 1 m ) log ⁡ 2 , ∼ ∼ ∼ P ω -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 η ( ω 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 ∈ N is a heavy-tailed branching process in random environment satisfying 𝔼[log⁡(E ω Z 1)] = ∞, in contrast to the case η ( ω 0 = 1 m ) = 0 studied by Nakashima (2013), where 𝔼[log⁡(E ω Z 1)] < ∞. 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 ∈ N is of independent interest.

. This article is devoted to studying the class of backward stochastic differential equations with delayed generators. We suppose the terminal value and the generator to be L p -integrable with p > 1. We derive a new type of estimation related to this BSDE. Next, we establish the existence and uniqueness result in two ways. First, we adapt an approximation technique used by Briand et al. (Stochastic Process. Appl. 108 (2003) 109-129) to the delayed BSDEs. Next, we use the Picard iterative procedure and revisit the result of Dos Reis et al. (Stochastic Process. Appl. 121 (9) (2011) 2114-2150), simplifying the proof and giving an explicit existence and uniqueness condition related to the Lipschitz constant K and the terminal time T.

Cramér’s type moderate deviations quantify the relative error of the tail probability approximation, and provide a criterion for whether the limiting tail probability can be used to estimate the tail probability under study. In this article, a Cramér moderate deviation theorem is established for random walks conditioned to stay positive in the sense of h-transform, which gives the relative error of the CLT proved by Bryn-Jones and Doney [J. Lond. Math. Soc., 2006]. Our approach is based on the strong approximation between random walks and Brownian motion.

. Consider a d-type supercritical branching process ( Z n i ) n ≥ 0 := ( Z n i ( 1 ) , ⋯ , Z n i ( d ) ) n ≥ 0 in an independent and identically distributed random environment ξ = ( ξ 0 , ξ 1 , ⋯ ) starting with one initial particle of type i. In this article, we prove an a.s. convergence rate for the associated non-negative martingale ( W n i ) n ≥ 0 introduced in Grama et al. [4 c4]

. We study a De Finetti’s optimal dividend and capital injection problem under a Markov additive model. The surplus process without dividend and capital injection is assumed to follow a spectrally positive Markov additive process (MAP). Dividend payments are made at the jump times of an independent Poisson process and capitals are injected to avoid bankruptcy. The aim of the paper is to characterize an optimal dividend and capital injection strategy that maximizes the expected total discounted dividends subtracted by the total discounted costs of capital injection. Applying the fluctuation and excursion theory for Lévy processes and the stochastic control theory, we first address an auxiliary dividend and capital injection control problem with a terminal payoff under the spectrally positive Lévy model. Using results obtained for this auxiliary problem and a fixed-point argument for iterations induced by the dynamic program, we characterize the optimal strategy of our prime control problem as a regime-modulated double-barrier Poissonian-continuous-reflection dividend and capital injection strategy. Besides, a numerical example is provided to illustrate the features of the optimal strategies. The impacts of model parameters are also studied.

This article aims to derive the Laplace-Stieltjes transform matrix for the total increment of a one-level process during the first passage of another level process to level zero in so-called the two-dimensional Markov modulated Brownian motion. The process comprises an irreducible continuous-time Markov process with a finite state space, alongside two level processes modulated by the Markov process. These paired level processes can be viewed as a two-dimensional Brownian motion, with Brownian parameters varying based on the Markov process. Due to the infeasibility of explicit computation, we formulate a nonsymmetric algebraic Riccati equation with a minimal nonnegative solution that represents the transform matrix through a matrix exponential function. To our knowledge, this achievement is innovative within the context of the two- dimensional Markov modulated Brownian motion.

Several generalizations of the classical Matérn type I and II hard-core point processes have been proposed in the literature during the last decades; for such processes, explicit results for the first- and second-order characteristics are available. We define here general thinning rules, both deterministic and probabilistic, of an inhomogeneous marked Poisson point process, and we provide explicit expressions for the factorial moment measures of any order of the thinned process and of its marginal process. As a byproduct, general expressions for the intensity measure and for the pair correlation function are given, recovering known results in the literature as special cases. In particular, we provide a general expression for the void probability function of the thinned process, from which we deduce upper and lower bounds. Possible applications in the study of the mean volume density of particular birth-and-growth models in materials science are discussed.

. This article considers a bidimensional delay-claim renewal risk model with stochastic returns and Brownian perturbations, in which the main claim and corresponding delayed claim from the same business line are dependent according to a general dependence structure, the price process of stochastic return is described as a geometric Lévy process, and the generic inter-arrival times of the two kinds of main claims are arbitrarily dependent. In the presence of subexponential or the intersection of dominatedly varying tailed and long-tailed main and delayed claims, the corresponding asymptotic formulas for four types of finite-time ruin probabilities are established, and some numerical studies are conducted to verify the accuracy of our asymptotic formulas.