Stochastic Processes and their Applications最新文献

In this work, percolation properties of device-to-device (D2D) networks in urban environments are investigated. On a street system modeled by a Poisson–Delaunay triangulation (PDT), users are of two types: given either by a Cox process supported by the edges of the PDT (i.e. on streets) or by a Bernoulli process on the vertices of the PDT (i.e. at crossroads). We state several percolation regimes for the resulting connectivity graph according to the parameters of the model. This work completes and specifies results of Le Gall et al. (2021). To do it, we take advantage of a percolation tool, inspired by enhancement techniques, used to our knowledge for the first time in the context of communication networks.

We consider a parametric version of the UST (Uniform Spanning Tree) measure on arbitrary directed weighted finite graphs with tuning (killing) parameter $q>0$. This is obtained by considering the standard random weighted spanning tree on the extended graph built by adding a ghost state $†$ and directed edges to it, of constant weights $q$, from any vertex of the original graph. The resulting measure corresponds to a random spanning rooted forest of the graph where the parameter $q$ tunes the intensity of the number of trees as follows: partitions with many trees are favored for $q>1$, while as $q\to 0$, the standard UST of the graph is recovered. We are interested in the behavior of the induced random partition, referred to as Loop-erased partitioning, which gives a correlated cluster model, as the multiscale parameter $q\in [0,\infty )$ varies.

Emergence of giant clusters in this correlated percolation model as a function of $q$ has been recently explored on certain dense growing graphs Avena et al. (2022). Herein we derive two types of results. First, we explore monotonicity properties in $q$ of this forest measure on general graphs showing in particular some counter-intuitive subtleties in non-reversible settings where the electrical-network interpretation of the UST observables gets partially lost. Second, by analyzing 2-points correlations on trees and various very sparse growing graph models, we characterize emerging macroscopic clusters, as $q$ scales with the graph size, and derive related phase diagrams.

We study the sample-path moderate deviation principle (MDP) for shot noise processes in the high intensity regime. The shot noise processes have a renewal arrival process, non-stationary noises (with arrival-time dependent distributions) and a general shot response function of the noises. The rate function in the MDP exhibits a memory phenomenon in this asymptotic regime, which is in contrast with that in the conventional time–space scaling regime. To prove the sample-path MDP, we first establish that this is equivalent to establishing the sample-path MDP of another process that is easier to study. We prove its finite-dimensional MDP and then establish the exponential tightness under the Skorohod $J_1$ topology. This results in the sample-path MDP in $D$ under the Skorohod $J_1$ topology with a rate function that is derived from the rate function in the finite-dimensional MDP using the tools of reproducing kernel Hilbert space. In the proofs, because of the non-stationarity of shot noise process, we establish a new exponential maximal inequality and use it to prove exponential tightness and the aforementioned equivalence.

The Lie groups $SU\left(2\right)$ and $SL(2,R)$ can be viewed as model spaces in subRiemannian geometry. Coupling two subelliptic Brownian motions on $SU\left(2\right)$ (resp. $SL(2,R)$) consists in simultaneously coupling two Brownian motions on the sphere (resp. the hyperbolic plane) and their swept areas. Using this approach we propose an explicit construction of a co-adapted successful coupling on $SU\left(2\right)$. The strategy is to alternate between reflection and synchronous (with noise) couplings on the sphere. We also describe some more general constructions of co-adapted couplings on $SU\left(2\right)$ and on $SL(2,R)$.

We study a stochastic control/stopping problem with a series of inequality-type and equality-type expectation constraints in a general non-Markovian framework. We demonstrate that the stochastic control/stopping problem with expectation constraints (CSEC) is independent of a specific probability setting and is equivalent to the constrained stochastic control/stopping problem in weak formulation (an optimization over joint laws of Brownian motion, state dynamics, diffusion controls and stopping rules on an enlarged canonical space). Using a martingale-problem formulation of controlled SDEs in spirit of Stroock and Varadhan (2006), we characterize the probability classes in weak formulation by countably many actions of canonical processes, and thus obtain the upper semi-analyticity of the CSEC value function. Then we employ a measurable selection argument to establish a dynamic programming principle (DPP) in weak formulation for the CSEC value function, in which the conditional expected costs act as additional states for constraint levels at the intermediate horizon.

This article extends (El Karoui and Tan, 2013) to the expectation-constraint case. We extend our previous work (Bayraktar and Yao, 2024) to the more complicated setting where the diffusion is controlled. Compared to that paper the topological properties of diffusion-control spaces and the corresponding measurability are more technically involved which complicate the arguments especially for the measurable selection for the super-solution side of DPP in the weak formulation.

To model bio-chemical reaction systems with diffusion one can either use stochastic, microscopic reaction–diffusion master equations or deterministic, macroscopic reaction–diffusion system. The connection between these two models is not only theoretically important but also plays an essential role in applications. This paper considers the macroscopic limits of the chemical reaction–diffusion master equation for first-order chemical reaction systems in highly heterogeneous environments. More precisely, the diffusion coefficients as well as the reaction rates are spatially inhomogeneous and the reaction rates may also be discontinuous. By carefully discretizing these heterogeneities within a reaction–diffusion master equation model, we show that in the limit we recover the macroscopic reaction–diffusion system with inhomogeneous diffusion and reaction rates.

A crinkled subordinator is an ${\ell }^2$-valued random process which can be thought of as a version of the usual one-dimensional subordinator with each out of countably many jumps being in a direction orthogonal to the directions of all other jumps. We show that the path of a $d$-dimensional random walk with $n$ independent identically distributed steps with heavy-tailed distribution of the radial components and asymptotically orthogonal angular components converges in distribution in the Hausdorff distance up to isometry and also in the Gromov–Hausdorff sense, if viewed as a random metric space, to the closed range of a crinkled subordinator, as $d,n\to \infty$.

Statistical inference for stochastic processes based on high frequency observations has been an active research area for more than two decades. One of the most well-known and widely studied problems has been the estimation of the quadratic variation of the continuous component of an Itô semimartingale with jumps. Several rate- and variance-efficient estimators have been proposed in the literature when the jump component is of bounded variation. However, to date, very few methods can deal with jumps of unbounded variation. By developing new high-order expansions of the truncated moments of a locally stable Lévy process, we propose a new rate- and variance-efficient volatility estimator for a class of Itô semimartingales whose jumps behave locally like those of a stable Lévy process with Blumenthal–Getoor index $Y\in (1,8/5)$ (hence, of unbounded variation). The proposed method is based on a two-step debiasing procedure for the truncated realized quadratic variation of the process and can also cover the case $Y<1$. Our Monte Carlo experiments indicate that the method outperforms other efficient alternatives in the literature in the setting covered by our theoretical framework.

We develop theory leading to testing procedures for the presence of a change point in the intraday volatility pattern. The new theory is developed in the framework of Functional Data Analysis. It is based on a model akin to the stochastic volatility model for scalar point-to-point returns. In our context, we study intraday curves, one curve per trading day. After postulating a suitable model for such functional data, we present three tests focusing, respectively, on changes in the shape, the magnitude and arbitrary changes in the sequences of the curves of interest. We justify the respective procedures by showing that they have asymptotically correct size and by deriving consistency rates for all tests. These rates involve the sample size (the number of trading days) and the grid size (the number of observations per day). We also derive the corresponding change point estimators and their consistency rates. All procedures are additionally validated by a simulation study and an application to US stocks.

We consider a finite collection of reinforced stochastic processes with a general network-based interaction among them. We provide sufficient and necessary conditions for the emergence of some form of almost sure asymptotic synchronization. Specifically, we identify three regimes: the first involves complete synchronization, where all processes converge towards the same random variable; the second exhibits almost sure convergence of the system, but no form of synchronization subsists; and the third reveals a scenario where there is almost sure asymptotic synchronization within the cyclic classes of the interaction matrix, together with an asymptotic periodic behavior among these classes.