Journal of Theoretical Probability最新文献

In this paper, we consider random variables and stochastic processes from the space ({textbf{F}}_psi (Omega )) and study approximation problems for such processes. The method of series decomposition of a stochastic process from ({textbf{F}}_psi (Omega )) is used to find an approximating process called a model. The rate of convergence of the model to the process in the uniform norm is investigated. We develop an approach for estimating the cut-off level of the model under given accuracy and reliability of the simulation.

Abstract

The probabilistic symbol is defined as the right-hand side derivative at time zero of the characteristic functions corresponding to the one-dimensional marginals of a time-homogeneous stochastic process. As described in various contributions to this topic, the symbol contains crucial information concerning the process. When leaving time-homogeneity behind, a modification of the symbol by inserting a time component is needed. In the present article, we show the existence of such a time-dependent symbol for non-homogeneous Itô processes. Moreover, for this class of processes, we derive maximal inequalities which we apply to generalize the Blumenthal–Getoor indices to the non-homogeneous case. These are utilized to derive several properties regarding the paths of the process, including the asymptotic behavior of the sample paths, the existence of exponential moments and the finiteness of p-variationa. In contrast to many situations where non-homogeneous Markov processes are involved, the space-time process cannot be utilized when considering maximal inequalities.

The almost-sure sample path behavior of the operator fractional Brownian motion with exponent D, including multivariate fractional Brownian motion, is investigated. In particular, the global and the local moduli of continuity of the sample paths are established. These results show that the global and the local moduli of continuity of the sample paths are completely determined by the real parts of the eigenvalues of the exponent D, as well as the covariance matrix at some unit vector. These results are applicable to multivariate fractional Brownian motion.

We give rates of convergence in the almost sure invariance principle for sums of dependent random variables with semi-exponential tails, whose coupling coefficients decrease at a sub-exponential rate. We show that the rates in the strong invariance principle are in powers of (log n). We apply our results to iid products of random matrices.

In this paper, we investigate the sojourn times of conditionally Gaussian processes, i.e., the sojourns of (xi (t)+lambda -zeta ,t^beta ) and (xi (t)(lambda -zeta ,t^beta )), (tin [0, T], T>0), where (xi ) is a Gaussian zero-mean stationary process and (lambda ) and (zeta ) are random variables independent of (xi (cdot )), and (beta >0) is a constant.

In this paper, we consider a large class of super-Brownian motions in ({mathbb {R}}) with spatially dependent branching mechanisms. We establish the almost sure growth rate of the mass located outside a time-dependent interval ((-delta t,delta t)) for (delta >0). The growth rate is given in terms of the principal eigenvalue (lambda _{1}) of the Schrödinger-type operator associated with the branching mechanism. From this result, we see the existence of phase transition for the growth order at (delta =sqrt{lambda _{1}/2}). We further show that the super-Brownian motion shifted by (sqrt{lambda _{1}/2},t) converges in distribution to a random measure with random density mixed by a martingale limit.

In this paper, the reflected McKean–Vlasov diffusion ov a convex domain is studied. We first establish the well-posedness of a coupled system of nonlinear stochastic differential equations via a fixed point theorem which is similar to that for partial differential equations. Moreover, the reason why we make different assumptions on drift and cross terms is given. Then, the propagation of chaos for the particle system is also obtained.

Based on the resolvent decomposition theorems presented very recently by Chen (J Theor Probab 33:2089–2118, 2020), in this paper we focus on investigating the fundamental problems of existence and uniqueness criteria for Denumerable Markov Processes with finitely many instantaneous states. Some elegant sufficient and necessary conditions are obtained for this less-investigated topic. A few important examples including the generalized Kolmogorov models are presented to illustrate our general results.

Given an intertwining relation between two finite Markov chains, we investigate how it can be transformed by conditioning the primal Markov chain to stay in a proper subset. A natural assumption on the underlying link kernel is put forward. The three classical examples of discrete Pitman, top-to-random shuffle and absorbed birth-and-death chain intertwinings serve as illustrations.