Moderate deviations for dynamic model governed by stationary process (Q2726269)

It is well known that a family \(X^{n}=(X_{t}^{n})\) of diffusion processes determined by the equation NEWLINE\[NEWLINEdX^{n}=a(X^{n})dt+(1/\sqrt{n^{k}})b(X^{n})dW_{t}NEWLINE\]NEWLINE subject to fixed initial point \(X_{0},\) where \(W_{t}\) is a Wiener process and \(k>0,\) obeys the large deviation principle (LDP) in the space of continuous functions supplied by the locally uniform metric. This LDP is characterized by the rate of speed \((1/n^{k})\) and some rate function. The author considers the same problem from more ``realistic'' model in which the Wiener process is replaced by its approximation \(W^{n}_{t}=(1/\sqrt{n})\int_{0}^{tn}\eta_{s} ds,\) where \(\eta_{t}\) is a stationary ergodic process, \(E\eta_{0}=0,\) satisfying some weak dependence conditions. Thus, \(X_{t}^{n}\) is defined now by an ordinary differential equation NEWLINE\[NEWLINEdX_{t}^{n}/dt=a(X^{n})+b(X^{n})(n/\sqrt{n^{k+1}})\eta_{tn}.NEWLINE\]NEWLINE Here \(\eta_{t}\) is the stationary process with the Wold decomposition \(\eta_{t}=\int_{-\infty}^{t}h(t-s) dN_{s}\) with respect to a process \(N_{t}\) with homogeneous independent square integrable increments and \(h(t)\) being some deterministic function. It is an example of the moderate deviations principle (MDP) for the family \(X^{n}_{t}.\) Instead of LDP it is used the abbreviation MDP to emphasize the fact that the rate function is of a diffusion type and is independent of \(k.\)NEWLINENEWLINEFor the entire collection see [Zbl 0956.00022].