### Abstract

In this paper we investigate the small time behavior of solutions of the Zakai equation. We derive a wave equation-like stochastic partial differential equation which is related to the Zakai equation. We are able to solve this equation for sufficiently smooth signals, and (approximately) transform these into solutions of the Zakai equation. We construct a Hadamardtype expansion for solutions of this partial differential equation and show how this expansion is related to a small time expansion of solutions of the Zakai equation.

Original language | English (US) |
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Pages (from-to) | 283-303 |

Number of pages | 21 |

Journal | Mathematical Systems Theory |

Volume | 20 |

Issue number | 1 |

DOIs | |

State | Published - Dec 1987 |

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### ASJC Scopus subject areas

- Theoretical Computer Science
- Mathematics(all)
- Computational Theory and Mathematics

### Cite this

**On the small time behavior of the nonlinear estimation problem for finite bandwidth signals.** / Taylor, Thomas.

Research output: Contribution to journal › Article

*Mathematical Systems Theory*, vol. 20, no. 1, pp. 283-303. https://doi.org/10.1007/BF01692071

}

TY - JOUR

T1 - On the small time behavior of the nonlinear estimation problem for finite bandwidth signals

AU - Taylor, Thomas

PY - 1987/12

Y1 - 1987/12

N2 - In this paper we investigate the small time behavior of solutions of the Zakai equation. We derive a wave equation-like stochastic partial differential equation which is related to the Zakai equation. We are able to solve this equation for sufficiently smooth signals, and (approximately) transform these into solutions of the Zakai equation. We construct a Hadamardtype expansion for solutions of this partial differential equation and show how this expansion is related to a small time expansion of solutions of the Zakai equation.

AB - In this paper we investigate the small time behavior of solutions of the Zakai equation. We derive a wave equation-like stochastic partial differential equation which is related to the Zakai equation. We are able to solve this equation for sufficiently smooth signals, and (approximately) transform these into solutions of the Zakai equation. We construct a Hadamardtype expansion for solutions of this partial differential equation and show how this expansion is related to a small time expansion of solutions of the Zakai equation.

UR - http://www.scopus.com/inward/record.url?scp=0010166768&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0010166768&partnerID=8YFLogxK

U2 - 10.1007/BF01692071

DO - 10.1007/BF01692071

M3 - Article

VL - 20

SP - 283

EP - 303

JO - Theory of Computing Systems

JF - Theory of Computing Systems

SN - 1432-4350

IS - 1

ER -