Stochastic Processes and their Applications最新文献

In this paper, we obtain a comparison theorem and an invariant representation theorem for backward stochastic differential equations (BSDEs) without any assumption on the second variable $z$. Using the two results, we further develop the theory of $g$-expectations. Filtration-consistent nonlinear expectation ($F$-expectation) provides an ideal characterization for the dynamical risk measures, asset pricing and utilities. We propose two new conditions: an absolutely continuous condition and a (locally Lipschitz) domination condition. Under the two conditions respectively, we prove that any $F$-expectation can be represented as a $g$-expectation. Our results contain a representation theorem for $n$-dimensional $F$-expectations in the Lipschitz case, and two representation theorems for 1-dimensional $F$-expectations in the locally Lipschitz case, which contain quadratic $F$-expectations.

In this paper, we first establish general bounds on the Fisher information distance to the class of normal distributions of Malliavin differentiable random variables. We then study the rate of Fisher information convergence in the central limit theorem for the solution of small noise stochastic differential equations and its additive functionals. We also show that the convergence rate is of optimal order.

We establish the existence of infinitely many global and stationary solutions in $C(R,C^{\vartheta })$ space for some $\vartheta >0$ to the three-dimensional Euler equations driven by an additive stochastic forcing. The result is based on a new stochastic version of the convex integration method, incorporating the stochastic convex integration method developed in Hofmanová et al. (2022) and pathwise estimates to derive uniform moment estimates independent of time.

Optimal transport (OT) based data analysis is often faced with the issue that the underlying cost function is (partially) unknown. This is addressed in this paper with the derivation of distributional limits for the empirical OT value when the cost function and the measures are estimated from data. For statistical inference purposes, but also from the viewpoint of a stability analysis, understanding the fluctuation of such quantities is paramount. Our results find direct application in the problem of goodness-of-fit testing for group families, in machine learning applications where invariant transport costs arise, in the problem of estimating the distance between mixtures of distributions, and for the analysis of empirical sliced OT quantities.

The established distributional limits assume either weak convergence of the cost process in uniform norm or that the cost is determined by an optimization problem of the OT value over a fixed parameter space. For the first setting we rely on careful lower and upper bounds for the OT value in terms of the measures and the cost in conjunction with a Skorokhod representation. The second setting is based on a functional delta method for the OT value process over the parameter space. The proof techniques might be of independent interest.

We analyze the non-equilibrium fluctuations of the partial symmetric simple exclusion process, SEP($\alpha$), which allows at most $\alpha \in N$ particles per site, and we put it in contact with stochastic reservoirs whose strength is regulated by a parameter $\theta \in R$. Setting $\alpha =1$, we find the results of Landim et al. (2008), Franco et al. (2019) and Gonçalves et al. (2020) and extend the known results to cover all range of $\theta$.

This paper studies the equilibrium consumption under external habit formation in a large population of agents. We first formulate problems under two types of conventional habit formation preferences, namely linear and multiplicative external habit formation, in a mean field game framework. In a log-normal market model with the asset specialization, we characterize one mean field equilibrium in analytical form in each problem, allowing us to understand some quantitative properties of the equilibrium strategy and conclude some financial implications caused by consumption habits from a mean-field perspective. In each problem with $n$ agents, we construct an approximate Nash equilibrium for the $n$-player game using the obtained mean field equilibrium when $n$ is sufficiently large. The explicit convergence order in each problem can also be obtained.

A polarized social network is modeled as a system of interacting marked point processes with memory of variable length. Each point process indicates the successive times in which a social actor expresses a “favorable” or “contrary” opinion. After expressing an opinion, the social pressure on the actor is reset to 0, waiting for the group’s reaction. The orientation and the rate at which an actor expresses an opinion is influenced by the social pressure exerted on it, modulated by a polarization coefficient. We prove that the network reaches an instantaneous but metastable consensus, when the polarization coefficient diverges.

We are interested in the invasion phase for stochastic processes with interactions. A single mutant with positive fitness arrives in a large resident population at equilibrium. By a now classical approach, the first stage of the invasion is well approximated by a branching process. The macroscopic phase, when the mutant population is of the same order as the resident population, is described by the limiting dynamical system. We capture the intermediate mesoscopic phase for the invasive population and obtain sharp approximations. It allows us to describe the fluctuations of the hitting times of thresholds, which inherit a large variance from the first stage. We apply our results to two models which are original motivations. In particular, we quantify the hitting times of critical values in cancer emergence and epidemics.

We study quasi-stationary distribution of the continuous-state branching process with competition introduced by Berestycki et al. (2018). This process is defined as the unique strong solution to a stochastic integral equation with jumps. An important example is the logistic branching process proposed by Lambert (2005). We establish the strong Feller property, trajectory Feller property, Lyapunov condition, weak Feller property and irreducibility, respectively. These properties together allow us to prove that if the competition is strong enough near $+\infty$, then there is a unique quasi-stationary distribution, which attracts all initial distributions with exponential rates.