Nonlinear filtering problem with contamination
This paper concerns a nonstandard filtering problem \((X^\varepsilon_t, Y_t^\varepsilon)_{t\geq 0}\) where \(X^\varepsilon_t\) is the unobserved signal and \(Y^\varepsilon_t\) is the observation. As \(\varepsilon\) goes to 0, the limit problem \((\overline X_t, \overline Y_t)\) is standard: Markovian diffusion process, or Gaussian diffusion. Let \(\overline\pi_t\) be the classical filtering estimate: \(\overline\pi_t (Y)= E[u (\overline X_t)/ \overline Y_s,s \leq t]|_{\overline Y_s=Y}\). It is used to approximate \(E[u (X^\varepsilon_t)/ Y^\varepsilon_s, s\leq t]= \pi^\varepsilon_t (Y^\varepsilon)\). Let \(Q^\varepsilon_T\), \(\overline Q_T\), \(\overline Q^\varepsilon_T\) denote the distributions of the processes \((X^\varepsilon_t, Y^\varepsilon_t)_{t\leq T}\), \((\overline X_t, \overline Y_t)_{t\leq T}\) and \((\overline X_t, Y_t^\varepsilon)_{t\leq T}\), respectively; if the total variation norm \(\| Q^\varepsilon_T- \overline Q_T\|\) goes to 0 as \(\varepsilon\) goes to 0, or: if \(\sup_{t\leq T} | X^\varepsilon_t -\overline X_t |\) goes to 0 in probability as \(\varepsilon\) goes to 0 and the total variation norm \(\|\overline Q^\varepsilon_T- \overline Q_T\|\) goes to 0 as \(\varepsilon\) goes to 0, then, for a suitable measurable function \(u\), it yields: \[ \lim_{\varepsilon \to 0} E\bigl[u (X^\varepsilon_t) -\overline \pi_t(Y^\varepsilon) \bigr]^2 <+\infty, \quad \lim_{ \varepsilon \to 0} E\bigl[\pi^\varepsilon_t (Y^\varepsilon)-\overline \pi(Y^\varepsilon) \bigr]^2 =0 , \] \[ \lim_{\varepsilon\to 0} E\bigl[u(X_t^\varepsilon)-\overline \pi_t(Y^\varepsilon) \bigr]^2=E \bigl[u (\overline X_t)- \overline\pi_t (\overline Y)\bigr]^2, \lim_{\varepsilon\to 0} E\bigl[u (X^\varepsilon_t) -\pi_t(Y^\varepsilon) \bigr]^2= E\bigl[u(\overline X_t)- \overline\pi_t (\overline Y) \bigr]^2. \] Two models satisfying such assumptions are presented. Some ``contamination'' terms are introduced in drift terms of the dynamics of \(X\) and \(Y\), destroying the Markov property. The noises in the two dynamics are independent.
scientific article; zbMATH DE number 1121386
scientific article