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Kalman-Bucy Filter in Continuous Time
Týbl, Ondřej ; Maslowski, Bohdan (advisor) ; Čoupek, Petr (referee)
In the Thesis we study the problem of linear filtration of Gaussian signals in finite-dimensional space. We use the Kalman-type equations for the filter to show that the filter depends continuously on the signal. Secondly, we show the same continuity property for the covariance of the error and verify existence and uniqueness of a solution to an integral equation that is satisfied by the filter even under more general assumptions. We present several examples of application of the continuity property that are based on the theory of stochastic differential equations driven by fractional Brownian motion. 1
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Filtering for Stochastic Evolution Equations
Kubelka, Vít ; Maslowski, Bohdan (advisor)
Filtering for Stochastic Evolution Equations Vít Kubelka Doctoral thesis Abstract Linear filtering problem for infinite-dimensional Gaussian processes is studied, the observation process being finite-dimensional. Integral equations for the filter and for covariance of the error are derived. General results are applied to linear SPDEs driven by Gauss-Volterra process observed at finitely many points of the domain and to delayed SPDEs driven by white noise. Subsequently, the continuous dependence of the filter and observation error on parameters which may be present both in the signal and the obser- vation process is proved. These results are applied to signals governed by stochastic heat equations driven by distributed or pointwise fractional noise. The observation process may be a noisy observation of the signal at given points in the domain, the position of which may depend on the parameter. 1
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Filtering for Stochastic Evolution Equations
Kubelka, Vít ; Maslowski, Bohdan (advisor)
Filtering for Stochastic Evolution Equations Vít Kubelka Doctoral thesis Abstract Linear filtering problem for infinite-dimensional Gaussian processes is studied, the observation process being finite-dimensional. Integral equations for the filter and for covariance of the error are derived. General results are applied to linear SPDEs driven by Gauss-Volterra process observed at finitely many points of the domain and to delayed SPDEs driven by white noise. Subsequently, the continuous dependence of the filter and observation error on parameters which may be present both in the signal and the obser- vation process is proved. These results are applied to signals governed by stochastic heat equations driven by distributed or pointwise fractional noise. The observation process may be a noisy observation of the signal at given points in the domain, the position of which may depend on the parameter. 1
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Filtering for Stochastic Evolution Equations
Kubelka, Vít ; Maslowski, Bohdan (advisor) ; Tudor, Ciprian (referee) ; Klebanov, Lev (referee)
Filtering for Stochastic Evolution Equations Vít Kubelka Doctoral thesis Abstract Linear filtering problem for infinite-dimensional Gaussian processes is studied, the observation process being finite-dimensional. Integral equations for the filter and for covariance of the error are derived. General results are applied to linear SPDEs driven by Gauss-Volterra process observed at finitely many points of the domain and to delayed SPDEs driven by white noise. Subsequently, the continuous dependence of the filter and observation error on parameters which may be present both in the signal and the obser- vation process is proved. These results are applied to signals governed by stochastic heat equations driven by distributed or pointwise fractional noise. The observation process may be a noisy observation of the signal at given points in the domain, the position of which may depend on the parameter. 1
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Kalman-Bucy Filter in Continuous Time
Týbl, Ondřej ; Maslowski, Bohdan (advisor) ; Čoupek, Petr (referee)
In the Thesis we study the problem of linear filtration of Gaussian signals in finite-dimensional space. We use the Kalman-type equations for the filter to show that the filter depends continuously on the signal. Secondly, we show the same continuity property for the covariance of the error and verify existence and uniqueness of a solution to an integral equation that is satisfied by the filter even under more general assumptions. We present several examples of application of the continuity property that are based on the theory of stochastic differential equations driven by fractional Brownian motion. 1
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