On large deviations in nonlinear filtering theory (Q2773388)

Let \(X^{\varepsilon}\) be the solution of NEWLINE\[NEWLINEX_t^{\varepsilon}=x+\sum_{i=1}^r\int_0^t \sigma_i(X_s^{\varepsilon}) dW_s^i + \varepsilon \sum_{j=1}^l \int_0^t\widetilde{\sigma}_j(X_s^{\varepsilon}) d\widetilde{W}_s^j+ \int_0^tb(X_s^{\varepsilon})ds,NEWLINE\]NEWLINE where \((C_0([0,1],R^r), W, P)\) and \((C_0([0,1], R^l), \widetilde{W}, \widetilde{P})\) are independent Wiener processes. Also let \(\varphi^{\varepsilon}\Phi= E\varphi(X^{\varepsilon}),\) where \(E\) is expectation w.r.t. \(P\) and \(\Phi\) is a \(C^2\)-function of the appropriate real spaces. Large deviations principles are proved for \((X^{\varepsilon}, \varepsilon >0)\) and for \((\varphi^{\varepsilon}\Phi\), \(\varepsilon >0)\), and an application to nonlinear filtering is discussed.