## Abstract

The smoothing of diffusions dx_{t} = f(x_{t}) dt + σ(x_{t}) dw_{t}, measured by a noisy sensor dy_{t} = h(x_{t}) dt + dv_{t}, where w_{t} and v_{t} are independent Wiener processes, is considered in this paper. By focussing our attention on the joint p.d.f. of (x_{τ} x_{t}), 0 ≤ τ < t, conditioned on the observation path {y_{s}, 0 ≤ s ≤ t}, the smoothing problem is represented as a solution of an appropriate joint filtering problem of the process, together with its random initial conditions. The filtering problem thus obtained possesses a solution represented by a Zakai-type forward equation. This solution of the smoothing problem differs from the common approach where, by concentrating on the conditional p.d.f. of x_{τ} alone, a set of 'forward and reverse' equations needs to be solved.

Original language | English |
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Pages (from-to) | 317-321 |

Number of pages | 5 |

Journal | Systems and Control Letters |

Volume | 7 |

Issue number | 4 |

DOIs | |

State | Published - Jul 1986 |

## Keywords

- Finite-dimensional filters
- Nonlinear filtering
- Smoothing