A Wong–Zakai approximation for random slow manifolds with application to parameter estimation

Abstract

We study a Wong-Zakai approximation for the random slow manifold of a slow-fast stochastic dynamical system. We first deduce the existence of the random slow manifold about an approximation system driven by an integrated Ornstein-Uhlenbeck (O-U) process. Then we compute the slow manifold of the approximation system, in order to gain insights of the long time dynamics of the original stochastic system. By restricting this approximation system to its slow manifold, we thus get a reduced slow random system. This reduced slow random system is used to accurately estimate a system parameter of the original system. An example is presented to illustrate this approximation.An Ornstein-Uhlenbeck (O-U) process is introduced historically to depict the velocity of the particle in Brownian motion. Its integration, the integrated O-U process, is regarded as the displacement of the particle [43]. The O-U process has a frequency-dependent power spectral density. Due to this characteristic behavior of its power spectral density, it is also called a colored noise to be distinguished from white noise [35]. When the correlation of collisions between the Brownian particle and the surrounding liquid molecules leads to a situation where the finite correlation time becomes important, the system driven by colored noise instead of white noise deserves investigation [35]. Numerous realistic models [8,27,35], analytical works [9, 28] and numerical simulations [33,17] concern the system driven by colored noise. When the correlation time tends to vanish, the colored noise will approach the white noise. According to the Wong-Zakai theorem [45,44], the system driven by colored noise (or integrated O-U process) will converge to the system driven by white noise (or Brownian motion). Inheriting the benefits of the general Wong-Zakai approximation, the system driven by integrated O-U process is a random differential equation (RDE). The RDE can be regarded as the ordinary differential equation (ODE) pathwisely. And it can be represented by deterministic Riemann integral which is more robust to approximate than stochastic integral in view of their definition. Thus it is easier to simulate a RDE than a stochastic differential equation (SDE) [11,36]. But differing from the piecewise linear approximation to Brownian motion [6], the integrated O-U process has a continuous derivative. This analytic property makes the system easier to analyze to some extent [2]. Due to these characteristics, it is meaningful to study both the dynamical behavior of system driven by integrated O-U process and the approximation property when the correlation time tends to infinity. In fact, Wong-Zakai theorem is extended to other situations [5,6,22,23,25,29,42]. The dimension of the state space has grown from one to finite, and then infinity. That means the state space can be a general Hilbert space. The driving process is extended from Brownian motion to semimartingales. Various modes of convergence are considered, such as convergence in the mean square, in probability, and al...