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Filtering with a limiter (improved performance). (English) Zbl 0931.60029

At times \(t_k= k\Delta\), one observes the sequence \((Y_0= 0)\) \(Y_{t_k}- Y_{t_{k-1}}= \mu X_{t_{k-1}}\Delta+ \sqrt\Delta B_k\), where 1. \(X\) solves \[ X(\omega, t)= X(\omega, 0)+ \alpha \int^t_0 X(\omega, s) ds+ \beta W(\omega, t), \] \(X(\omega,0)\) a Gaussian random variable and \(W\) a standard Wiener process independent of \(\{B_k\}\),
2. \(\{B_k\}\) is a sequence of independent, identically distributed random variables with \(E[B_k]= 0\), \(V[B_k]= \sigma^2_B\),
3. \(\mu\), \(\alpha\), \(\beta\), and \(\sigma^2_B\) are known parameters.
Intuitively, the sequence \(\{Y_{t_k}\}\) represents discrete observations of the Itô process \[ Y(\omega, t)= \mu\int^t_0 X(\omega,s) ds+\sigma_BB(\omega, t), \] where \(B\) is a standard Wiener process independent of \(X\). It can in fact be shown that \(Y\) is the limit, in the appropriate sense, as \(\Delta\) goes to \(0\), of the process \(Y^\Delta\) defined by the relationship: \[ Y^\Delta(\omega, t)= \sum^\infty_{k=1} I_{[t_{k-1}, t_k[}(t) Y_{t_{k-1}}(\omega). \] It turns out that \(\widehat X(\omega,t)= E[X(\cdot,t)\mid \sigma_t(Y)]\) has a functional representation in the form of \(\widehat X(\omega, t)= \Phi(Y(\omega,\cdot), t)\), where \(\Phi\) can be computed for arguments in \(D[\mathbb{R}_+]\). So one can set \(\widehat X^\Delta(\omega, t)= \Phi(Y^\Delta(\omega,\cdot), t)\), for which \[ \lim_{\Delta\to 0} E[(X(\cdot, t)- \widehat X^\Delta(\cdot, t))^2]= P(t)= E[(X(\cdot, t)- \widehat X(\cdot,t))^2]. \] Of course, in “practice”, one can only hope to compute \(\widetilde X^\Delta(\omega, t)= E[X(\cdot, t)\mid \sigma_t(Y^\Delta)]\). But, if one assumes that \(B_1\) has a density \(f_B\) with finite Fisher information, there are two closely related Itô processes, \(Z\) and \(\widehat Z\), such that \(\widetilde X^\Delta(\cdot, t)\) converges in law, as \(\Delta\) goes to \(0\), to \[ E[X(\cdot, t)\mid \sigma_t(Z)\vee \sigma_t(\widetilde Z)]\equiv E[X(\cdot, t)\mid \sigma_t(\widetilde Z)]. \] One then has that \[ \lim_{\Delta\to 0} E[(X(\cdot, t)- \widetilde X^\Delta(\cdot, t))^2]= \widetilde P(t)= E[(X(\cdot, t)- E[X(\cdot, t)\mid \sigma_t(\widetilde Z)])^2], \] and furthermore that \(P(t)>\widetilde P(t)\), \(t>0\). The aim of the paper is to provide a filter for which \(\widetilde P(t)\) is asymptotically attainable. That requires the following assumptions. Set \(\varphi_B= -f_B'/f_B\). \(f_B\) must be twice continuously differentiable, \(\varphi_B'\) must be well defined and satisfy a linear growth condition, and finally one must have \(E[\varphi^2_B(B_1)]< \infty\)! The results of some simulations for \(f_B\) a normal mixture are exhibited.

MSC:

60G35 Signal detection and filtering (aspects of stochastic processes)
93E11 Filtering in stochastic control theory
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