In this manuscript, we address open questions raised by Dieker and Yakir (2014), who proposed a novel method of estimating (discrete) Pickands constants $\mathcal{H}^\delta_\alpha$ using a family of estimators $\xi^\delta_\alpha(T)$, $T>0$, where $\alpha\in(0,2]$ is the Hurst parameter, and $\delta\geq0$ is the step size of the regular discretization grid. We derive an upper bound for the discretization error $\mathcal{H}_\alpha^0 - \mathcal{H}_\alpha^\delta$, whose rate of convergence agrees with Conjecture 1 of Dieker and Yakir (2014) in the case $\alpha\in(0,1]$ and agrees up to logarithmic terms for $\alpha\in(1,2)$. Moreover, we show that all moments of $\xi_\alpha^\delta(T)$ are uniformly bounded and the bias of the estimator decays no slower than $\exp\{-\mathcal CT^{\alpha}\}$, as T becomes large.

We study the tail asymptotics of two functionals (the maximum and the sum of the marks) of a generic cluster in two sub-models of the marked Poisson cluster process, namely the renewal Poisson cluster process and the Hawkes process. Under the hypothesis that the governing components of the processes are regularly varying, we extend results due to [6, 19], notably relying on Karamata’s Tauberian Theorem to do so. We use these asymptotics to derive precise large-deviation results in the fashion of [32] for the just-mentioned processes.

We consider the stochastic volatility model obtained by adding a compound Hawkes process to the volatility of the well-known Heston model. A Hawkes process is a self-exciting counting process with many applications in mathematical finance, insurance, epidemiology, seismology, and other fields. We prove a general result on the existence of a family of equivalent (local) martingale measures. We apply this result to a particular example where the sizes of the jumps are exponentially distributed. Finally, a practical application to efficient computation of exposures is discussed.

Given a sequence of independent random vectors taking values in ${\mathbb R}^d$ and having common continuous distribution function F, say that the $n^{\rm \scriptsize}$th observation sets a (Pareto) record if it is not dominated (in every coordinate) by any preceding observation. Let $p_n(F) \equiv p_{n, d}(F)$ denote the probability that the $n^{\rm \scriptsize}$th observation sets a record. There are many interesting questions to address concerning pn and multivariate records more generally, but this short paper focuses on how pn varies with F, particularly if, under F, the coordinates exhibit negative dependence or positive dependence (rather than independence, a more-studied case). We introduce new notions of negative and positive dependence ideally suited for such a study, called negative record-setting probability dependence (NRPD) and positive record-setting probability dependence (PRPD), relate these notions to existing notions of dependence, and for fixed $d \geq 2$ and $n \geq 1$ prove that the image of the mapping pn on the domain of NRPD (respectively, PRPD) distributions is $[p^*_n, 1]$ (resp., $[n^{-1}, p^*_n]$), where $p^*_n$ is the record-setting probability for any continuous F governing independent coordinates.

It is proved that for families of stochastic operators on a countable tensor product, depending smoothly on parameters, any spectral projection persists smoothly, where smoothness is defined using norms based on ideas of Dobrushin. A rigorous perturbation theory for families of stochastic operators with spectral gap is thereby created. It is illustrated by deriving an effective slow two-state dynamics for a three-state probabilistic cellular automaton.

We introduce a bivariate tempered space-fractional Poisson process (BTSFPP) by time-changing the bivariate Poisson process with an independent tempered $\alpha$-stable subordinator. We study its distributional properties and its connection to differential equations. The Lévy measure for the BTSFPP is also derived. A bivariate competing risks and shock model based on the BTSFPP for predicting the failure times of items that undergo two random shocks is also explored. The system is supposed to break when the sum of two types of shock reaches a certain random threshold. Various results related to reliability, such as reliability function, hazard rates, failure density, and the probability that failure occurs due to a certain type of shock, are studied. We show that for a general Lévy subordinator, the failure time of the system is exponentially distributed with mean depending on the Laplace exponent of the Lévy subordinator when the threshold has a geometric distribution. Some special cases and several typical examples are also demonstrated.

We define the co-spectral radius of inclusions ${\mathcal S}\leq {\mathcal R}$ of discrete, probability- measure-preserving equivalence relations as the sampling exponent of a generating random walk on the ambient relation. The co-spectral radius is analogous to the spectral radius for random walks on $G/H$ for inclusion $H\leq G$ of groups. For the proof, we develop a more general version of the 2–3 method we used in another work on the growth of unimodular random rooted trees. We use this method to show that the walk growth exists for an arbitrary unimodular random rooted graph of bounded degree. We also investigate how the co-spectral radius behaves for hyperfinite relations, and discuss new critical exponents for percolation that can be defined using the co-spectral radius.

Qu, Dassios, and Zhao (2021) suggested an exact simulation method for tempered stable Ornstein–Uhlenbeck processes, but their algorithms contain some errors. This short note aims to correct their algorithms and conduct some numerical experiments.

We investigate different geometrical properties, related to Carleson measures and pseudo-hyperbolic separation, of inhomogeneous Poisson point processes on the unit disk. In particular, we give conditions so that these random sequences are almost surely interpolating for the Hardy, Bloch or weighted Dirichlet spaces.

In this paper we consider the filtering of partially observed multidimensional diffusion processes that are observed regularly at discrete times. This is a challenging problem which requires the use of advanced numerical schemes based upon time-discretization of the diffusion process and then the application of particle filters. Perhaps the state-of-the-art method for moderate-dimensional problems is the multilevel particle filter of Jasra et al. (SIAM J. Numer. Anal. 55 (2017), 3068–3096). This is a method that combines multilevel Monte Carlo and particle filters. The approach in that article is based intrinsically upon an Euler discretization method. We develop a new particle filter based upon the antithetic truncated Milstein scheme of Giles and Szpruch (Ann. Appl. Prob. 24 (2014), 1585–1620). We show empirically for a class of diffusion problems that, for $\epsilon>0$ given, the cost to produce a mean squared error (MSE) of $\mathcal{O}(\epsilon^2)$ in the estimation of the filter is $\mathcal{O}(\epsilon^{-2}\log(\epsilon)^2)$. In the case of multidimensional diffusions with non-constant diffusion coefficient, the method of Jasra et al. (2017) requires a cost of $\mathcal{O}(\epsilon^{-2.5})$ to achieve the same MSE.

We give an extension of Lê’s stochastic sewing lemma. The stochastic sewing lemma proves convergence in $L_m$ of Riemann type sums $\sum _{[s,t] \in \pi } A_{s,t}$ for an adapted two-parameter stochastic process A, under certain conditions on the moments of $A_{s,t}$ and of conditional expectations of $A_{s,t}$ given $\mathcal F_s$. Our extension replaces the conditional expectation given $\mathcal F_s$ by that given $\mathcal F_v$ for $v<s$, and it allows to make use of asymptotic decorrelation properties between $A_{s,t}$ and $\mathcal F_v$ by including a singularity in $(s-v)$. We provide three applications for which Lê’s stochastic sewing lemma seems to be insufficient. The first is to prove the convergence of Itô or Stratonovich approximations of stochastic integrals along fractional Brownian motions under low regularity assumptions. The second is to obtain new representations of local times of fractional Brownian motions via discretization. The third is to improve a regularity assumption on the diffusion coefficient of a stochastic differential equation driven by a fractional Brownian motion for pathwise uniqueness and strong existence.

Expectiles have received increasing attention as a risk measure in risk management because of their coherency and elicitability at the level $\alpha\geq1/2$. With a view to practical risk assessments, this paper delves into the worst-case expectile, where only partial information on the underlying distribution is available and there is no closed-form representation. We explore the asymptotic behavior of the worst-case expectile on two specified ambiguity sets: one is through the Wasserstein distance from a reference distribution and transforms this problem into a convex optimization problem via the well-known Kusuoka representation, and the other is induced by higher moment constraints. We obtain precise results in some special cases; nevertheless, there are no unified closed-form solutions. We aim to fully characterize the extreme behaviors; that is, we pursue an approximate solution as the level $\alpha $ tends to 1, which is aesthetically pleasing. As an application of our technique, we investigate the ambiguity set induced by higher moment conditions. Finally, we compare our worst-case expectile approach with a more conservative method based on stochastic order, which is referred to as ‘model aggregation’.

In this paper we extend results on reconstruction of probabilistic supports of independent and identically distributed random variables to supports of dependent stationary ${\mathbb R}^d$-valued random variables. All supports are assumed to be compact of positive reach in Euclidean space. Our main results involve the study of the convergence in the Hausdorff sense of a cloud of stationary dependent random vectors to their common support. A novel topological reconstruction result is stated, and a number of illustrative examples are presented. The example of the Möbius Markov chain on the circle is treated at the end with simulations.

We investigate an optimal stopping problem for the expected value of a discounted payoff on a regime-switching geometric Brownian motion under two constraints on the possible stopping times: only at exogenous random times, and only during a specific regime. The main objectives are to show that an optimal stopping time exists as a threshold type and to derive expressions for the value functions and the optimal threshold. To this end, we solve the corresponding variational inequality and show that its solution coincides with the value functions. Some numerical results are also introduced. Furthermore, we investigate some asymptotic behaviors.

We present a detailed analysis of random motions moving in higher spaces with a natural number of velocities. In the case of the so-called minimal random dynamics, under some broad assumptions, we give the joint distribution of the position of the motion (for both the inner part and the boundary of the support) and the number of displacements performed with each velocity. Explicit results for cyclic and complete motions are derived. We establish useful relationships between motions moving in different spaces, and we derive the form of the distribution of the movements in arbitrary dimension. Finally, we investigate further properties for stochastic motions governed by non-homogeneous Poisson processes.

We consider the problem of obtaining effective representations for the solutions of linear, vector-valued stochastic differential equations (SDEs) driven by non-Gaussian pure-jump Lévy processes, and we show how such representations lead to efficient simulation methods. The processes considered constitute a broad class of models that find application across the physical and biological sciences, mathematics, finance, and engineering. Motivated by important relevant problems in statistical inference, we derive new, generalised shot-noise simulation methods whenever a normal variance-mean (NVM) mixture representation exists for the driving Lévy process, including the generalised hyperbolic, normal-gamma, and normal tempered stable cases. Simple, explicit conditions are identified for the convergence of the residual of a truncated shot-noise representation to a Brownian motion in the case of the pure Lévy process, and to a Brownian-driven SDE in the case of the Lévy-driven SDE. These results provide Gaussian approximations to the small jumps of the process under the NVM representation. The resulting representations are of particular importance in state inference and parameter estimation for Lévy-driven SDE models, since the resulting conditionally Gaussian structures can be readily incorporated into latent variable inference methods such as Markov chain Monte Carlo, expectation-maximisation, and sequential Monte Carlo.

We show that for arrival processes, the ‘harmonic new better than used in expectation’ (HNBUE) (or ‘harmonic new worse than used in expectation’, HNWUE) property is a sufficient condition for inequalities between the time and customer averages of the system if the state of the system between arrival epochs is stochastically decreasing and convex and the lack of anticipation assumption is satisfied. HNB(W)UE is a wider class than NB(W)UE, being the largest of all available classes of distributions with positive (negative) aging properties. Thus, this result represents an important step beyond existing result on inequalities between time and customer averages, which states that for arrival processes, the NB(W)UE property is a sufficient condition for inequalities.

This paper considers the first passage times to constant boundaries and the two-sided exit problem for Lévy process with a characteristic exponent in which at least one of the two jumps having rational Laplace transforms. The joint distribution of the first passage times and undershoot/overshoot are obtained. The processes recover many models that have appeared in the literature such as the compound Poisson risk models, the perturbed compound Poisson risk models, and their dual ones. As applications, we obtain the solutions for popular path-dependent options such as lookback and barrier options in terms of Laplace transforms.

The term moderate deviations is often used in the literature to mean a class of large deviation principles that, in some sense, fills the gap between a convergence in probability of some random variables to a constant, and a weak convergence to a centered Gaussian distribution (when such random variables are properly centered and rescaled). We talk about noncentral moderate deviations when the weak convergence is towards a non-Gaussian distribution. In this paper we prove a noncentral moderate deviation result for the bivariate sequence of sums and maxima of independent and identically distributed random variables bounded from above. We also prove a result where the random variables are not bounded from above, and the maxima are suitably normalized. Finally, we prove a moderate deviation result for sums of partial minima of independent and identically distributed exponential random variables.

Centrality measures aim to indicate who is important in a network. Various notions of ‘being important’ give rise to different centrality measures. In this paper, we study how important the central vertices are for the connectivity structure of the network, by investigating how the removal of the most central vertices affects the number of connected components and the size of the giant component. We use local convergence techniques to identify the limiting number of connected components for locally converging graphs and centrality measures that depend on the vertex’s neighbourhood. For the size of the giant, we prove a general upper bound. For the matching lower bound, we specialise to the case of degree centrality on one of the most popular models in network science, the configuration model, for which we show that removal of the highest-degree vertices destroys the giant most.