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.

In this paper, we consider random dynamical systems formed by concatenating maps acting on the unit interval $[0,1]$ in an independent and identically distributed (i.i.d.) fashion. Considered as a stationary Markov process, the random dynamical system possesses a unique stationary measure $\nu $. We consider a class of non-square-integrable observables $\phi $, mostly of form $\phi (x)=d(x,x_0)^{-{1}/{\alpha }}$, where $x_0$ is a non-recurrent point (in particular a non-periodic point) satisfying some other genericity conditions and, more generally, regularly varying observables with index $\alpha \in (0,2)$. The two types of maps we concatenate are a class of piecewise $C^2$ expanding maps and a class of intermittent maps possessing an indifferent fixed point at the origin. Under conditions on the dynamics and $\alpha $, we establish Poisson limit laws, convergence of scaled Birkhoff sums to a stable limit law, and functional stable limit laws in both the annealed and quenched case. The scaling constants for the limit laws for almost every quenched realization are the same as those of the annealed case and determined by $\nu $. This is in contrast to the scalings in quenched central limit theorems where the centering constants depend in a critical way upon the realization and are not the same for almost every realization.

We prove that the local time of random walks conditioned to stay positive converges to the corresponding local time of three-dimensional Bessel processes by proper scaling. Our proof is based on Tanaka’s pathwise construction for conditioned random walks and the derivation of asymptotics for mixed moments of the local time.

We consider Gaussian approximation in a variant of the classical Johnson–Mehl birth–growth model with random growth speed. Seeds appear randomly in $\mathbb{R}^d$ at random times and start growing instantaneously in all directions with a random speed. The locations, birth times, and growth speeds of the seeds are given by a Poisson process. Under suitable conditions on the random growth speed, the time distribution, and a weight function $h\;:\;\mathbb{R}^d \times [0,\infty) \to [0,\infty)$, we prove a Gaussian convergence of the sum of the weights at the exposed points, which are those seeds in the model that are not covered at the time of their birth. Such models have previously been considered, albeit with fixed growth speed. Moreover, using recent results on stabilization regions, we provide non-asymptotic bounds on the distance between the normalized sum of weights and a standard Gaussian random variable in the Wasserstein and Kolmogorov metrics.

We consider the propagation of a stochastic SIR-type epidemic in two connected populations: a relatively small local population of interest which is surrounded by a much larger external population. External infectives can temporarily enter the small population and contribute to the spread of the infection inside this population. The rules for entry of infectives into the small population as well as their length of stay are modeled by a general Markov queueing system. Our main objective is to determine the distribution of the total number of infections within both populations. To do this, the approach we propose consists of deriving a family of martingales for the joint epidemic processes and applying classical stopping time or convergence theorems. The study then focuses on several particular cases where the external infection is described by a linear branching process and the entry of external infectives obeys certain specific rules. Some of the results obtained are illustrated by numerical examples.

This paper examines the issue of derivative pricing within the framework of a fractional stochastic volatility model. We present a deterministic partial differential equation system to derive an approximate expression for the derivative price. The proposed approach allows for the stochastic volatility to be expressed as a composition of deterministic functions of time and a fractional Ornstein–Uhlenbeck process. We apply this method to the European option pricing under the fractional Stein–Stein volatility model, demonstrating its feasibility and reliability through numerical simulations. Our numerical simulations also illustrate the impact of the parameters in the fractional stochastic volatility model on the option price.