Journal of Theoretical Probability最新文献

Consider a generalized time-dependent Pólya urn process defined as follows. Let (din mathbb {N}) be the number of urns/colors. At each time n, we distribute (sigma _n) balls randomly to the d urns, proportionally to f, where f is a valid reinforcement function. We consider a general class of positive reinforcement functions (mathcal {R}) assuming some monotonicity and growth condition. The class (mathcal {R}) includes convex functions and the classical case (f(x)=x^{alpha }), (alpha >1). The novelty of the paper lies in extending stochastic approximation techniques to the d-dimensional case and proving that eventually the process will fixate at some random urn and the other urns will not receive any balls anymore.

This article is concerned with the limiting behavior of invariant measures for stochastic reaction–diffusion equations driven by nonlinear noise on unbounded thin domains. We first show the existence of invariant measures when the diffusion terms are globally Lipschitz continuous. The uniform estimates on the tails of solutions are employed to present the tightness of a family of probability distributions of solutions in order to overcome the non-compactness of usual Sobolev embeddings on unbounded domains. Then, we prove any limit of invariant measures of the equations defined on ((n+1))-dimensional unbounded thin domains must be an invariant measure of the limiting system as the thin domains collapse onto the space (mathbb {R}^n).

Consider a finite undirected graph and place an urn with balls of two colours at each vertex. At every discrete time step, for each urn, a fixed number of balls are drawn from that same urn with probability p and from a randomly chosen neighbour of that urn with probability (1-p). Based on what is drawn, the urns then reinforce themselves or their neighbours. For every ball of a given colour in the sample, in case of Pólya-type reinforcement, a constant multiple of balls of that colour is added while in case of Friedman-type reinforcement, balls of the other colour are reinforced. These different choices for reinforcement give rise to multiple models. In this paper, we study the convergence of the fraction of balls of either colour across urns for all of these models. We show that in most cases the urns synchronize, that is, the fraction of balls of either colour in each urn converges to the same limit almost surely. A different kind of asymptotic behaviour is observed on bipartite graphs. We also prove similar results for the case of finite directed graphs.

We deduce stability and pathwise uniqueness for a McKean–Vlasov equation with random coefficients and a multidimensional Brownian motion as driver. Our analysis focuses on a non-Lipschitz continuous drift and includes moment estimates for random Itô processes that are of independent interest. For deterministic coefficients, we provide unique strong solutions even if the drift fails to be of affine growth. The theory that we develop rests on Itô’s formula and leads to pth moment and pathwise exponential stability for (pge 2) with explicit Lyapunov exponents.

We consider a Galton–Watson tree where each node is marked independently of each other with a probability depending on its out-degree. Using a penalization method, we exhibit new martingales where the number of marks up to level (n-1) appears. Then, we use these martingales to define new probability measures via a Girsanov transformation and describe the distribution of the random trees under these new probabilities.

We obtain Berry–Esseen-type bounds for the sum of random variables with a dependency graph and uniformly bounded moments of order (delta in (2,infty ]) using a Fourier transform approach. Our bounds improve the state-of-the-art obtained by Stein’s method in the regime where the degree of the dependency graph is large.

In this paper, we consider the composition of a homogeneous Poisson process with an independent time-fractional Poisson process. We call this composition the generalized iterated Poisson process (GIPP). The probability law in terms of the fractional Bell polynomials, governing fractional differential equations, and the compound representation of the GIPP are obtained. We give explicit expressions for mean and covariance and study the long-range dependence property of the GIPP. It is also shown that the GIPP is over-dispersed. Some results related to first-passage time distribution and the hitting probability are also examined. We define the compound and the multivariate versions of the GIPP and explore their main characteristics. Further, we consider a surplus model based on the compound version of the iterated Poisson process (IPP) and derive several results related to ruin theory. Its applications using the Poisson–Lindley and the zero-truncated geometric distributions are also provided. Finally, simulated sample paths for the IPP and the GIPP are presented.

In the present paper, we study the existence and optimal controllability of a multi-term time-fractional stochastic system with non-instantaneous impulses. Using semigroup theory, stochastic techniques, and Krasnoselskii’s fixed point theorem, we first establish the existence of a mild solution. Further, we obtain that there exists an optimal state-control pair for the system under certain assumptions. Some examples are given to illustrate the abstract results.

The (L^p) maximal inequalities for martingales are one of the classical results in the theory of stochastic processes. Here, we establish the sharp moderate maximal inequalities for one-dimensional diffusion processes, which generalize the (L^p) maximal inequalities for diffusions. Moreover, we apply our theory to many specific examples, including the Ornstein–Uhlenbeck (OU) process, Brownian motion with drift, reflected Brownian motion with drift, Cox–Ingersoll–Ross process, radial OU process, and Bessel process. The results are further applied to establish the moderate maximal inequalities for some high-dimensional processes, including the complex OU process and general conformal local martingales.

Well-posedness is derived for singular path-distribution dependent stochastic differential equations (SDEs) with non-degenerate noise, where the drift is allowed to be singular in the current state, but maintains local Lipschitz continuity in the historical path, and the coefficients are Lipschitz continuous with respect to a weighted variation distance in the distribution variable. Notably, this result is new even for classical path-dependent SDEs where the coefficients are distribution independent. Moreover, by strengthening the local Lipschitz continuity to Lipschitz continuity and replacing the weighted variation distance with the Wasserstein distance, we also obtain well-posedness.