Examples of FCLT in Random Environment

Abstract

We consider a diffusion type process being a weak solution of Itô’s equation

$$dX^\varepsilon_t=b(\omega,X^\varepsilon_t/\varepsilon)dt+\sigma(\omega,X^\varepsilon_t/\varepsilon)dB_t$$

relative to fixed initial condition $X^{\varepsilon}_{0}=x_{0}$, Brownian motion (B _t ) _t≥0, and ergodic stationary random process (b(ω,u),σ(ω,u)) _u∈ℝ, treated as a “random environment”, where ε is a small positive parameter. Random environment and Brownian motions are independent random objects. Functions b(ω,u),σ(ω,u) are uniformly bounded and function σ ²(ω,u) is uniformly positive. Random environment obeys some “weak dependence” property (see (9)). We show that the family $\{(X^{\varepsilon}_{t})_{t\le T}\}_{\varepsilon \to0}$ converges in law to a continuous Gaussian process X=(X _t ) _t≤T with the expectation $\mathsf{E}X_{t}=x_{0}+\mathbf{b}t$ and covariance cov (X _t ,X _s )=a(t ∧ s), where

$$\mathbf{a}=1\Big/\mathsf{E}\frac{1}{\sigma^2(\omega,0)},\qquad \mathbf{b}=\mathsf{E}\frac{b(\omega,0)}{\sigma^2(\omega,0)}\Big /\mathsf{E}\frac{1}{\sigma^2(\omega,0)}.$$

The author gratefully acknowledges the anonymous referee who pointed out author’s attention on Papanicolau and Varadhan paper [8].