Stochastics and Dynamics

We consider white noise perturbations of a nonlinear dynamical system in the neighborhood of an unstable critical point with linearization given by a Jordan block of full dimension. For the associated exit problem, we study the joint limiting behavior of the exit location and exit time, in the vanishing noise limit. The exit typically happens near one of two special deterministic points associated with the eigendirection, and we obtain several more terms in the expansion for the exit point. The leading correction term is deterministic and logarithmic in the noise magnitude, while the random remainder satisfies a scaling limit.

In this paper, we study dynamical properties such as hypercyclicity, supercyclicity, frequent hypercyclicity and chaoticity for transition operators associated to countable irreducible Markov chains. As particular cases, we consider simple random walks on $\mathbb{Z}$ and ${\mathbb{Z}}_{+}$.

This paper is devoted to studying dimensional reduction for slow-fast data assimilation driven by Gaussian noise via stochastic averaging. We apply an energy method to show that the probability density for the reduced lower-dimensional system approximates that for the original system in mean square. In other words, the reduced system filter thus effectively captures the filter of the original system.

In this paper, we study a class of Anticipated Backward Stochastic Differential Equations (ABSDE) with jumps. The solution of the ABSDE is a triple $(Y,Z,\psi )$ where $Y$ is a semimartingale, and $(Z,\psi )$ are the diffusion and jump coefficients. We allow the driver of the ABSDE to have linear growth on the uniform norm of $Y$’s future paths, as well as quadratic and exponential growth on the spot values of $(Z,\psi )$, respectively. The existence of the unique solution is proved for Markovian and non-Markovian settings with different structural assumptions on the driver. In the former case, some regularities on $(Z,\psi )$ with respect to the forward process are also obtained.

In this paper, we consider the ergodicity and transience of the Cox–Ingersoll–Ross (CIR) interest rate model with Markov switching. Using the theory of $M$-matrices, we give some necessary and sufficient conditions for ergodicity of the CIR interest rate model with Markov switching. Besides, we show that the transition semigroup converges to the stationary distribution at an exponential rate in the Wasserstein distance. Finally, two examples are presented to illustrate our theory.

By using the Malliavin calculus, the Driver-type integration by parts formula is established for the semigroup associated to SPDEs with Multiplicative Noise. Moreover, estimates on the logarithmic derivative of the transition probability measure are obtained. A concrete example to describe evolution of spin systems on discrete lattices is given to illustrate our main result.

In this paper, we first review the penalization method for solving deterministic Skorokhod problems in non-convex domains and establish estimates for problems with $\alpha$-Hölder continuous functions. With the help of these results obtained previously for deterministic problems, we pathwisely define the reflected $G$-Brownian motion and prove its existence and uniqueness in a Banach space. Finally, multi-dimensional reflected stochastic differential equations driven by $G$-Brownian motion are investigated via a fixed-point argument.

In this paper, we consider the self-normalized asymptotic properties of the parameter estimators in the fractional Ornstein–Uhlenbeck process. The deviation inequalities, Cramér-type moderate deviations and Berry–Esseen bounds are obtained. The main methods include the deviation inequalities and moderate deviations for multiple Wiener–Itô integrals [P. Major, Tail behavior of multiple integrals and U-statistics, Probab. Surv. 2 (2005) 448–505; On a multivariate version of Bernsteins inequality, Electron. J. Probab. 12 (2007) 966–988; M. Schulte and C. Thäle, Cumulants on Wiener chaos: Moderate deviations and the fourth moment theorem, J. Funct. Anal. 270 (2016) 2223–2248], as well as the Delta methods in large deviations [F. Q. Gao and X. Q. Zhao, Delta method in large deviations and moderate deviations for estimators, Ann. Statist. 39 (2011) 1211–1240]. For applications, we propose two test statistics which can be used to construct confidence intervals and rejection regions in the hypothesis testing for the drift coefficient. It is shown that the Type II errors tend to be zero exponential when using the proposed test statistics.