Stochastic Processes and their Applications最新文献

In this paper, we establish an abstract framework for the approximation of the invariant probability measure for a Markov semigroup. Following Pagès and Panloup (2022) we use an Euler scheme with decreasing step (unadjusted Langevin algorithm). Under some contraction property with exponential rate and some regularization properties, we give an estimate of the error in total variation distance. This abstract framework covers the main results in Pagès and Panloup (2022) and Chen et al. (2023). As a specific application we study the convergence in total variation distance to the invariant measure for jump type equations. The main technical difficulty consists in proving the regularization properties — this is done under an ellipticity condition, using Malliavin calculus for jump processes.

The Malliavin differentiability of a SDE plays a crucial role in the study of density smoothness and ergodicity among others. For Gaussian driven SDEs the differentiability issue is solved essentially in Cass et al., (2013). In this paper, we consider the Malliavin differentiability for the Euler scheme of such SDEs. We will focus on SDEs driven by fractional Brownian motions (fBm), which is a very natural class of Gaussian processes. We derive a uniform (in the step size $n$) path-wise upper-bound estimate for the Euler scheme for stochastic differential equations driven by fBm with Hurst parameter $H>1/3$ and its Malliavin derivatives.

In this paper we introduce and study renewal–reward processes in random environments where each renewal involves a reward taking values in a Banach space. We derive quenched large deviation principles and identify the associated rate functions in terms of variational formulas involving correctors. We illustrate the theory with three examples: compound Poisson processes in random environments, pinning of polymers at interfaces with disorder, and returns of Markov chains in dynamic random environments.

It has been claimed in Aldous et al. (2004) that all Lévy trees are mixings of inhomogeneous continuum random trees. We give a rigorous proof of this claim in the case of a stable branching mechanism, relying on a new procedure for recovering the tree distance from the graphical spanning trees that works simultaneously for stable trees and inhomogeneous continuum random trees.

This paper shows the nonlinear stochastic Feynman–Kac formula holds under quadratic growth. For this, we initiate the study of backward doubly stochastic differential equations (BDSDEs, for short) with quadratic growth. The existence, uniqueness, and comparison theorem for one-dimensional BDSDEs are proved when the generator $f(t,Y,Z)$ grows in $Z$ quadratically and the terminal value is bounded, by introducing innovative approaches. Furthermore, in this framework, we utilize BDSDEs to provide a probabilistic representation of solutions to semilinear stochastic partial differential equations (SPDEs, for short) in Sobolev spaces, and use it to prove the existence and uniqueness of such SPDEs, thereby extending the nonlinear stochastic Feynman–Kac formula for linear growth introduced by Pardoux and Peng (1994).

We introduce and study a non-oriented first passage percolation model having a property of statistical invariance by time reversal. This model is defined in a graph having directed edges and the passage times associated with each set of outgoing edges from a given vertex are distributed according to a generalized Bernoulli–Exponential law and i.i.d. among vertices. We derive the statistical invariance property by time reversal through a zero-temperature limit of the random walk in Dirichlet environment model.

This note was motivated by natural questions related to oriented percolation on a layered environment that introduces long range dependence. As a convenient tool, we are led to deal with questions on the strict decrease of the percolation parameter in the oriented setup when an extra dimension is added.

A new model for general cyclical long memory is introduced, by means of random modulation of certain bivariate long memory time series. This construction essentially decouples the two key features of cyclical long memory: quasi-periodicity and long-term persistence. It further allows for a general cyclical phase in cyclical long memory time series. Several choices for suitable bivariate long memory series are discussed, including a parametric fractionally integrated vector ARMA model. The parametric models introduced in this work have explicit autocovariance functions that can be readily used in simulation, estimation, and other tasks.

In this article, we consider a generalized First-passage percolation model, where each edge in $Z^d$ is independently assigned an infinite weight with probability $1-p$, and a random finite weight otherwise. The existence and positivity of the time constant have been established in Cerf and Théret (2016). Recently, using sophisticated multi-scale renormalizations, Cerf and Dembin (2022) proved that the time constant of chemical distance in super-critical percolation is Lipschitz continuous. In this work, we propose a different approach leveraging lattice animal theory and a simple one-step renormalization with the aid of Russo’s formula, to show the Lipschitz continuity of the time constant in generalized First-passage percolation.

The distribution of a one-dimensional drifted Brownian motion conditioned on its first hitting time to 0 is the same as a three-dimensional Bessel bridge. By applying the time change in Lamperti’s relation to this result, Matsumoto and Yor (2001) showed a relation between Brownian motions with opposite drifts. In two subsequent papers (Matsumoto and Yor, 2000; 2001), they established a geometric lifting of the process 2M-B in Pitman’s theorem, known as the Matsumoto–Yor process. They also established an equality in law involving Inverse Gaussian distribution and its reciprocal (as processes), known as the Matsumoto–Yor property, by conditioning some exponential Wiener functional.

Sabot and Zeng (2020) generalized some results on drifted Brownian motion conditioned on its first hitting. More precisely, they introduced a family of Brownian semimartingales with interacting drifts, for which when conditioned on the vector $\tau$ (the hitting times to 0 of each component), their joint law is the same as for independent three-dimensional Bessel bridges. The distribution of $\tau$ is a generalization of Inverse Gaussian distribution in multi-dimension and it is related to a random potential $\beta$ that appears in the study of the Vertex Reinforced Jump Process.

The aim of this paper is to generalize the results of Matsumoto and Yor (2001, 2000) in the context of these interacting Brownian semimartingales. We apply a Lamperti-type time change to the previous family of interacting Brownian motions and we obtain a multi-dimensional opposite drift theorem. Moreover, we also give a multi-dimensional counterpart of the Matsumoto–Yor process and its intertwining relation with interacting geometric Brownian motions.