In this article, we investigate averaging principle for stochastic hyperbolicparabolic equations with two time-scales, in which both the slow and fast components are perturbed by multiplicative noises. Particularly, we prove that the rate of strong convergence for the slow component to the averaged dynamics is of order 1/2, which significantly improves the order 1/4 established in our previous work.
In this paper, we consider the problem of parameter estimation for reflected Ornstein-Uhlenbeck process with large drift. The classical trajectory fitting estimator is inconsistent when the drift coefficient θ < 0, whereas it is consistent when θ ≥ 0. To overcome this limitation, we introduce an improved trajectory fitting estimator, which remains consistent for all θ ∈ ℝ. The consistency and asymptotic distribution of the improved trajectory fitting estimator are established under various combinations of the sign of θ. Computer simulations are performed to illustrate our theory.
In this paper, we address the problem of absolute continuity of translated Rosenblatt measures on the path space. In [P. & Ccaron;oupek, P. K & rcaron;& iacute;& zcaron; and B. Maslowski, Parameter estimation and singularity of laws on the path space for SDEs driven by Rosenblatt processes, Stochastic Process. Appl. 179 (2025) 104499], it is shown that there is no probability measure that would be equivalent to the original probability measure and under which a Rosenblatt path with a linear drift would again be a Rosenblatt path. Here, we show that if the Rosenblatt path is shifted in a direction belonging to a class of nontrivial Gaussian variables (that consists of a deterministic shift and a Wiener integral with respect to a fractional Brownian motion with a related Hurst parameter), such a measure exists. We also give several examples to demonstrate the scope of the result.
This paper discusses the weak pullback attractors for a damped stochastic fractional Schr & ouml;dinger equation on & Ropf;(n) with n >= 2. By utilizing the stochastic Strichartz estimates and a stopping time technique argument, the existence and uniqueness of a global solution for the systems with the nonlinear term |u|(2 sigma)u are proven. Furthermore, we define a mean random dynamical system due to the uniqueness of the solution, which has a unique weak pullback mean random attractor in L-rho(Omega; L-2(& Ropf;(n))). This result highlights the long-term dynamics of a broad class of stochastic fractional dispersion equations.
