Journal of Theoretical Probability最新文献

In this paper, we consider doubly reflected backward stochastic differential equations driven by G-Brownian motion with uniformly continuous coefficients. The existence of solutions can be obtained by a monotone convergence argument, a linearization method, a penalization method and the method of Picard iteration.

In this work, we prove the joint convergence in distribution of q variables modulo one obtained as partial sums of a sequence of i.i.d. square-integrable random variables multiplied by a common factor given by some function of an empirical mean of the same sequence. The limit is uniformly distributed over ([0,1]^q). To deal with the coupling introduced by the common factor, we assume that the absolutely continuous (with respect to the Lebesgue measure) part of the joint distribution of the random variables is nonzero, so that the convergence in the central limit theorem for this sequence holds in total variation distance. While our result provides a generalization of Benford’s law to a data-adapted mantissa, our main motivation is the derivation of a central limit theorem for the stratified resampling mechanism, which is performed in the companion paper (Flenghi and Jourdain, Central limit theorem for the stratified selection mechanism, 2023, http://arxiv.org/abs/2308.02186).

The 6-vertex model holds significance in various mathematical and physical domains. The configurations of the 6-vertex model correspond to the paths in multigraphs. This article focuses on calculating the transition probability for the simple random walk on these multigraphs. An intriguing aspect of the findings is the utilization of continued fractions in the computation of the transition probability.

Let G be a locally compact Hausdorff group, and let P(G) denote the class of all regular probability measures on G. It is well known that P(G) forms a semigroup under the convolution of measures. In this paper, we prove that P(G) is not algebraically regular in the sense that not every element has a generalized inverse. Additionally, we attempt to identify algebraically regular elements in some exceptional cases. Several supporting examples are provided to justify these assumptions.

In this paper, we establish the functional large deviation principle (LDP) for the Kac–Stroock approximations of a wild class of Gaussian processes constructed by telegraph types of integrals with (L^2)-integrands under mild conditions and find the explicit form for their rate functions. Our investigation includes a broad range of kernels, such as those related to Brownian motions, fractional Brownian motions with whole Hurst parameters, and Ornstein–Uhlenbeck processes. Furthermore, we consider a class of non-Markovian stochastic differential equations driven by the Kac–Stroock approximation and establish their Freidlin–Wentzell type LDP. The rate function clearly indicates an interesting phase transition phenomenon as the approximating rate moves from one region to the other.

Takeda and Yano (Electron J Probab 28:1–35, 2023) determined the limit of Lévy processes conditioned to avoid zero via various random clocks in terms of Doob’s (h)-transform, where the limit processes may differ according to the choice of random clocks. The purpose of this paper is to investigate sample path behaviors of the limit processes in long time and in short time.

In this paper, we consider a generalized birth–death process (GBDP) and examine its linear versions. Using its transition probabilities, we obtain the system of differential equations that governs its state probabilities. The distribution function of its waiting time in state s given that it starts in state s is obtained. For a linear version of it, namely the generalized linear birth–death process (GLBDP), we obtain the probability generating function, mean, variance and the probability of ultimate extinction of population. Also, we obtain the maximum likelihood estimate of its parameters. The differential equations that govern the joint cumulant generating functions of the population size with cumulative births and cumulative deaths are derived. In the case of constant birth and death rates in GBDP, the explicit forms of the state probabilities, joint probability mass functions of population size with cumulative births and cumulative deaths, and their marginal probability mass functions are obtained. It is shown that the Laplace transform of an integral of GBDP satisfies its Kolmogorov backward equation with certain scaled parameters. The first two moments of the path integral of GLBDP are obtained. Also, we consider the immigration effect in GLBDP for two different cases. An application of a linear version of GBDP and its path integral to a vehicles parking management system is discussed. Later, we introduce a time-changed version of the GBDP where time is changed via an inverse stable subordinator. We show that its state probabilities are governed by a system of fractional differential equations.

Let ({mathbb {R}}^N_+= [0,infty )^N). We here make new contributions concerning a class of random fields ((X_t)_{tin {mathbb {R}}^N_+}) which are known as multiparameter Lévy processes. Related multiparameter semigroups of operators and their generators are represented as pseudo-differential operators. We also provide a Phillips formula concerning the composition of ((X_t)_{tin {mathbb {R}}^N_+}) by means of subordinator fields. We finally define the composition of ((X_t)_{tin {mathbb {R}}^N_+}) by means of the so-called inverse random fields, which gives rise to interesting long-range dependence properties. As a byproduct of our analysis, we present a model of anomalous diffusion in an anisotropic medium which extends the one treated in Beghin et al. (Stoch Proc Appl 130:6364–6387, 2020), by improving some of its shortcomings.

The threshold Ornstein–Uhlenbeck process is a stochastic process that satisfies a stochastic differential equation with a drift term modeled as a piecewise linear function and a diffusion term characterized by a positive constant. This paper addresses the challenge of determining both the number and values of thresholds based on the continuously observed process. We present three testing algorithms aimed at determining the unknown number and values of thresholds, in conjunction with least squares estimators for drift parameters. The limiting distribution of the proposed test statistic is derived. Additionally, we employ sequential and global methods to determine the values of thresholds, and prove their weak convergence. Monte Carlo simulation results are provided to illustrate and support our theoretical findings. We utilize the model to estimate the term structure of US treasury rates and currency foreign exchange rates.

Let N(t) be the collection of particles alive at time t in a branching Brownian motion in (mathbb {R}^d), and for (uin N(t)), let ({textbf{X}}_u(t)) be the position of particle u at time t. For (theta in mathbb {R}^d), we define the additive measures of the branching Brownian motion by

In this paper, under some conditions on the offspring distribution, we give asymptotic expansions of arbitrary order for (mu _t^theta (({textbf{a}}, {textbf{b}}])) and (mu _t^theta ((-infty , {textbf{a}}])) for (theta in mathbb {R}^d) with (Vert theta Vert <sqrt{2}), where ((textbf{a}, textbf{b}]:=(a_1, b_1]times cdots times (a_d, b_d]) and ((-infty , textbf{a}]:=(-infty , a_1]times cdots times (-infty , a_d]) for (textbf{a}=(a_1,cdots , a_d)) and (textbf{b}=(b_1,cdots , b_d)). These expansions sharpen the asymptotic results of Asmussen and Kaplan (Stoch Process Appl 4(1):1–13, 1976) and Kang (J Korean Math Soc 36(1): 139–157, 1999) and are analogs of the expansions in Gao and Liu (Sci China Math 64(12):2759–2774, 2021) and Révész et al. (J Appl Probab 42(4):1081–1094, 2005) for branching Wiener processes (a particular class of branching random walks) corresponding to (theta ={textbf{0}}).