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A 1-dimensional nonlinear filtering problem. (English) Zbl 1190.60030

The authors propose a solution to the nonlinear filtering problem for systems described by the following system and observation equations \[ \begin{aligned} dX_t&=(F(t)X_t+\alpha(t))dt+(C(t)X_t+\beta(t))dU_t,\quad F(t)\in\mathbb R,C(t)\in\mathbb R, \\ dZ_t&=(G(t)X_t+\gamma(t))dt+(D(t)X_t+\xi(t))dV_u, \quad G(t)\in\mathbb R,D(t)\in\mathbb R, \end{aligned} \] \(F,G,C,D\) are bounded on bounded intervals, \(\alpha(t)\in\mathbb R\), \(\beta(t)\in\mathbb R\), \(\gamma(t)\in\mathbb R\), \(\xi(t)\in\mathbb R\), \(Z_0=0\), \(X_0\) is normally distributed (and independent of \(\{U_t\},\{V_t\}\)). The corresponding stochastic differential equation and Riccati equation are derived which determine solution \(\hat{X}_t=E[X_t|{\mathcal G}_t]\) of the filtering problem.

MSC:

60G35 Signal detection and filtering (aspects of stochastic processes)
62M20 Inference from stochastic processes and prediction
93E11 Filtering in stochastic control theory
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References:

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