We establish a sample path moderate deviation principle for the integrated shot noise process with Poisson arrivals and non-stationary noises. As in Pang and Taqqu (2019), we assume that the noise is conditionally independent given the arrival times, and the distribution of each noise depends on its arrival time. As applications, we derive moderate deviation principles for the workload process and the running maximum process for a stochastic fluid queue with the integrated shot noise process as the input; we also show that a steady-state distribution exists and derive the exact tail asymptotics.

In a recent paper, the authors studied the distribution properties of a class of exchangeable processes, called measure-valued Pólya sequences (MVPSs), which arise as the observation process in a generalized urn sampling scheme. Here we present several results in the form of ‘sufficientness’ postulates that characterize their predictive distributions. In particular, we show that exchangeable MVPSs are the unique exchangeable models whose predictive distributions are a mixture of the marginal distribution and the average of a probability kernel evaluated at past observations. When the latter coincides with the empirical measure, we recover a well-known result for the exchangeable model with a Dirichlet process prior. In addition, we provide a ‘pure’ sufficientness postulate for exchangeable MVPSs that does not assume a particular analytic form for the predictive distributions. Two other sufficientness postulates consider the case when the state space is finite.

The Wiener–Hopf factors of a Lévy process are the maximum and the displacement from the maximum at an independent exponential time. The majority of explicit solutions assume the upward jumps to be either phase-type or to have a rational Laplace transform, in which case the traditional expressions are lengthy expansions in terms of roots located by means of Rouché’s theorem. As an alternative, compact matrix formulas are derived, with parameters computable by iteration schemes.

We consider the count of subgraphs with an arbitrary configuration of endpoints in the random-connection model based on a Poisson point process on ${\mathord{\mathbb R}}^d$. We present combinatorial expressions for the computation of the cumulants and moments of all orders of such subgraph counts, which allow us to estimate the growth of cumulants as the intensity of the underlying Poisson point process goes to infinity. As a consequence, we obtain a central limit theorem with explicit convergence rates under the Kolmogorov distance and connectivity bounds. Numerical examples are presented using a computer code in SageMath for the closed-form computation of cumulants of any order, for any type of connected subgraph, and for any configuration of endpoints in any dimension $d{\geq} 1$. In particular, graph connectivity estimates, Gram–Charlier expansions for density estimation, and correlation estimates for joint subgraph counting are obtained.

This paper defines and studies a broad class of shock models by assuming that a Markovian arrival process models the arrival pattern of shocks. Under the defined class, we show that the system’s lifetime follows the well-known phase-type distribution. Further, we examine the age replacement policy for systems with a continuous phase-type distribution, identifying sufficient conditions for determining the optimal replacement time. Since phase-type distributions are dense in the class of lifetime distributions, our findings for the age replacement policy are widely applicable. We include numerical examples and graphical illustrations to support our results.

In this note, we formulate a ‘one-sided’ version of Wormald’s differential equation method. In the standard ‘two-sided’ method, one is given a family of random variables that evolve over time and which satisfy some conditions, including a tight estimate of the expected change in each variable over one-time step. These estimates for the expected one-step changes suggest that the variables ought to be close to the solution of a certain system of differential equations, and the standard method concludes that this is indeed the case. We give a result for the case where instead of a tight estimate for each variable’s expected one-step change, we have only an upper bound. Our proof is very simple and is flexible enough that if we instead assume tight estimates on the variables, then we recover the conclusion of the standard differential equation method.

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.

We prove a large deviation principle for the slow-fast rough differential equations (RDEs) under the controlled rough path (RP) framework. The driver RPs are lifted from the mixed fractional Brownian motion (FBM) with Hurst parameter $H\in (1/3,1/2)$. Our approach is based on the continuity of the solution mapping and the variational framework for mixed FBM. By utilizing the variational representation, our problem is transformed into a qualitative property of the controlled system. In particular, the fast RDE coincides with Itô stochastic differential equation (SDE) almost surely, which possesses a unique invariant probability measure with frozen slow component. We then demonstrate the weak convergence of the controlled slow component by averaging with respect to the invariant measure of the fast equation and exploiting the continuity of the solution mapping.

Let $\{\omega _n\}_{n\geq 1}$ be a sequence of independent and identically distributed random variables on a probability space $(\Omega , \mathcal {F}, \mathbb {P})$, each uniformly distributed on the unit circle $\mathbb {T}$, and let $\ell _n=cn^{-\tau }$ for some $c>0$ and $0<\tau <1$. Let $I_{n}=(\omega _n,\omega _n+\ell _n)$ be the random interval with left endpoint $\omega _n$ and length $\ell _n$. We study the asymptotic property of the covering time $N_n(x)=\sharp \{1\leq k\leq n: x\in I_k\}$ for each $x\in \mathbb {T}$. We prove the quenched central limit theorem for the covering time, that is, $\mathbb {P}$-almost surely,

$$ \begin{align*}\frac{N_n(x)-\mathbb{E}_{\mathbb{P}}(N_n(x))}{\sqrt{\sum_{k=1}^n \ell_k(1-\ell_k)}}\end{align*} $$

converges in law to the standard normal distribution.

We study the last exit time that a spectrally negative Lévy process is below zero until it reaches a positive level b, denoted by $g_{\tau_b^+}$. We generalize the results of the infinite-horizon last exit time explored by Chiu and Yin (2005) by incorporating a random horizon $\tau_b^+$, which represents the first passage time above b. We derive an explicit expression for the joint Laplace transform of $g_{\tau_b^+}$ and $\tau_b^+$ by utilizing a hybrid observation scheme approach proposed by Li, Willmot, and Wong (2018). We further study the optimal prediction of $g_{\tau_b^+}$ in the $L_1$ sense, and find that the optimal stopping time is the first passage time above a level $y_b^{\ast}$, with an explicit characterization of the stopping boundary $y_b^{\ast}$. As examples, Brownian motion with drift and the Cramér–Lundberg model with exponential jumps are considered.

We consider stochastic differential equations (SDEs) driven by a fractional Brownian motion with a drift coefficient that is allowed to be arbitrarily close to criticality in a scaling sense. We develop a comprehensive solution theory that includes strong existence, path-by-path uniqueness, existence of a solution flow of diffeomorphisms, Malliavin differentiability and $\rho $-irregularity. As a consequence, we can also treat McKean-Vlasov, transport and continuity equations.

Let $G$ be a group. The notion of linear sofic approximations of $G$ over an arbitrary field $F$ was introduced and systematically studied by Arzhantseva and Păunescu [3]. Inspired by one of the results of [3], we introduce and study the invariant $\kappa _F(G)$ that captures the quality of linear sofic approximations of $G$ over $F$. In this work, we show that when $F$ has characteristic zero and $G$ is linear sofic over $F$, then $\kappa _F(G)$ takes values in the interval $[1/2,1]$ and $1/2$ cannot be replaced by any larger value. Further, we show that under the same conditions, $\kappa _F(G)=1$ when $G$ is torsion-free. These results answer a question posed by Arzhantseva and Păunescu [3] for fields of characteristic zero. One of the new ingredients of our proofs is an effective non-concentration estimates for random walks on finitely generated abelian groups, which may be of independent interest.

In this paper, the model of bisexual branching processes affected by viral infectivity and with random control functions in independent and identically distributed (i.i.d.) random environments is established and the Markov property is given firstly. Then the relations of the probability generating functions of this model are studied, and some sufficient conditions for process extinction under common mating functions are presented. Finally, the limiting behaviors of the considered model after proper normalization, such as the sufficient conditions for the convergence in L1 and L2 and almost everywhere convergence, are investigated under the condition that the random control functions are super additive.

We consider the extremes of the logarithm of the characteristic polynomial of matrices from the C$\beta $E ensemble. We prove convergence in distribution of the centered maxima (of the real and imaginary parts) toward the sum of a Gumbel variable and another independent variable, which we characterize as the total mass of a ‘derivative martingale’. We also provide a description of the landscape near extrema points.

We study a version of the stochastic control problem of minimizing the sum of running and controlling costs, where control opportunities are restricted to independent Poisson arrival times. Under a general setting driven by a general Lévy process, we show the optimality of a periodic barrier strategy, which moves the process upward to the barrier whenever it is observed to be below it. The convergence of the optimal solutions to those in the continuous-observation case is also shown.

Let $B^{H}$ be a d-dimensional fractional Brownian motion with Hurst index $H\in(0,1)$, $f\,:\,[0,1]\longrightarrow\mathbb{R}^{d}$ a Borel function, and $E\subset[0,1]$, $F\subset\mathbb{R}^{d}$ are given Borel sets. The focus of this paper is on hitting probabilities of the non-centered Gaussian process $B^{H}+f$. It aims to highlight how each component f, E and F is involved in determining the upper and lower bounds of $\mathbb{P}\{(B^H+f)(E)\cap F\neq \emptyset \}$. When F is a singleton and f is a general measurable drift, some new estimates are obtained for the last probability by means of suitable Hausdorff measure and capacity of the graph $Gr_E(f)$. As application we deal with the issue of polarity of points for $(B^H+f)\vert_E$ (the restriction of $B^H+f$ to the subset $E\subset (0,\infty)$).

Early investigation of Pólya urns considered drawing balls one at a time. In the last two decades, several authors have considered multiple drawing in each step, but mostly for schemes involving two colors. In this manuscript, we consider multiple drawing from urns of balls of multiple colors, formulating asymptotic theory for specific urn classes and addressing more applications. The class we consider is affine and tenable, built around a ‘core’ square matrix. We examine cases where the urn is irreducible and demonstrate its relationship to matrix irreducibility for its core matrix, with examples provided. An index for the drawing schema is derived from the eigenvalues of the core. We identify three regimes: small, critical, and large index. In the small-index regime, we find an asymptotic Gaussian law. In the critical-index regime, we also find an asymptotic Gaussian law, albeit with a difference in the scale factor, which involves logarithmic terms. In both of these regimes, we have explicit forms for the structure of the mean and the covariance matrix of the composition vector (both exact and asymptotic). In all three regimes we have strong laws.

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.

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.