Journal of Theoretical Probability最新文献

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}}).

By investigating the regularity of the nonlinear semigroup (P_t^*) associated with the distribution dependent second-order stochastic differential equations, the Harnack inequality is derived when the drift is Lipschitz continuous in the measure variable under the distance induced by the functions being (beta )-Hölder continuous (with (beta > frac{2}{3})) on the degenerate component and square root of Dini continuous on the non-degenerate one. The results extend the existing ones in which the drift is Lipschitz continuous in (L^2)-Wasserstein distance.

In this paper, we consider stochastic reaction–diffusion equations with superlinear drift on the real line (mathbb {R}) driven by space–time white noise. A Freidlin–Wentzell large deviation principle is established by a modified weak convergence method on the space (C([0,T], C_textrm{tem}(mathbb {R}))), where (C_textrm{tem}(mathbb {R}):={fin C(mathbb {R}): sup _{xin mathbb {R}} left( |f(x)|e^{-lambda |x|}right) <infty text { for any } lambda >0}). Obtaining the main result in this paper is challenging due to the setting of unbounded domain, the space–time white noise, and the superlinear drift term without dissipation. To overcome these difficulties, the specially designed family of norms on the Fréchet space (C([0,T], C_textrm{tem}(mathbb {R}))), one-order moment estimates of the stochastic convolution, and two nonlinear Gronwall-type inequalities play an important role.

We study the hitting probabilities of the solution to a system of d stochastic heat equations with additive noise subject to Dirichlet boundary conditions. We show that for any bounded Borel set with positive ((d-6))-dimensional capacity, the solution visits this set almost surely.

We study the density fluctuations at equilibrium of the multi-species stirring process, a natural multi-type generalization of the symmetric (partial) exclusion process. In the diffusive scaling limit, the resulting process is a system of infinite-dimensional Ornstein–Uhlenbeck processes that are coupled in the noise terms. This shows that at the level of equilibrium fluctuations the species start to interact, even though at the level of the hydrodynamic limit each species diffuses separately. We consider also a generalization to a multi-species stirring process with a linear reaction term arising from species mutation. The general techniques used in the proof are based on the Dynkin martingale approach, combined with duality for the computation of the covariances.

Here, we study the long-term behaviour of the non-explosion probability for continuous-state branching processes in a Lévy environment when the branching mechanism is given by the negative of the Laplace exponent of a subordinator. In order to do so, we study the law of this family of processes in the infinite mean case and provide necessary and sufficient conditions for the process to be conservative, i.e. that the process does not explode in finite time a.s. In addition, we establish precise rates for the non-explosion probabilities in the subcritical and critical regimes, first found by Palau et al. (ALEA Lat Am J Probab Math Stat 13(2):1235–1258, 2016) in the case when the branching mechanism is given by the negative of the Laplace exponent of a stable subordinator.

For any fixed real (a > 0) and (x in {mathbb {R}}^d, d ge 1), we consider the real-valued random process ((S_n)_{n ge 0}) defined by ( S_0= a, S_n= a+ln vert g_ncdots g_1xvert , n ge 1), where the (g_k, k ge 1, ) are i.i.d. nonnegative random matrices. By using the strategy initiated by Denisov and Wachtel to control fluctuations in cones of d-dimensional random walks, we obtain an asymptotic estimate and bounds on the probability that the process ((S_n)_{n ge 0}) remains nonnegative up to time n and simultaneously belongs to some compact set ([b, b+ell ]subset {mathbb {R}}^+_*) at time n.

This paper gives a new property for stochastic processes, called square-mean (mu -)pseudo-S-asymptotically Bloch-type periodicity. We show how this property is preserved under some operations, such as composition and convolution, for stochastic processes. Our main results extend the classical results on S-asymptotically Bloch-type periodic functions. We also apply our results to some problems involving semilinear stochastic integrodifferential equations in abstract spaces