scholarly journals The Kalman-Bucy filter for integrable Lévy processes with infinite second moment

2015 ◽  
Vol 52 (03) ◽  
pp. 636-648
Author(s):  
David Applebaum ◽  
Stefan Blackwood
Keyword(s):  
System Dynamics ◽  
Lévy Processes ◽  
Levy Processes ◽  
Infinite Variance ◽  
Second Moment ◽  
Finite Dimensional ◽  
Observation Noise

We extend the Kalman-Bucy filter to the case where both the system and observation processes are driven by finite dimensional Lévy processes, but whereas the process driving the system dynamics is square-integrable, that driving the observations is not; however it remains integrable. The main result is that the components of the observation noise that have infinite variance make no contribution to the filtering equations. The key technique used is approximation by processes having bounded jumps.

scholarly journals The Kalman-Bucy filter for integrable Lévy processes with infinite second moment

2015 ◽  
Vol 52 (3) ◽  
pp. 636-648 ◽  
Author(s):  
David Applebaum ◽  
Stefan Blackwood
Keyword(s):  
System Dynamics ◽  
Lévy Processes ◽  
Levy Processes ◽  
Infinite Variance ◽  
Second Moment ◽  
Finite Dimensional ◽  
Observation Noise

We extend the Kalman-Bucy filter to the case where both the system and observation processes are driven by finite dimensional Lévy processes, but whereas the process driving the system dynamics is square-integrable, that driving the observations is not; however it remains integrable. The main result is that the components of the observation noise that have infinite variance make no contribution to the filtering equations. The key technique used is approximation by processes having bounded jumps.

scholarly journals Application of Lévy processes in modelling (geodetic) time series with mixed spectra

2021 ◽  
Vol 28 (1) ◽  
pp. 121-134
Author(s):  
Jean-Philippe Montillet ◽  
Xiaoxing He ◽  
Kegen Yu ◽  
Changliang Xiong
Keyword(s):  
Time Series ◽  
Lévy Processes ◽  
Stable Process ◽  
Functional Model ◽  
Noise Spectrum ◽  
Levy Processes ◽  
Infinite Variance ◽  
Residual Time ◽  
First Case ◽  
Heavy Tailed

Abstract. Recently, various models have been developed, including the fractional Brownian motion (fBm), to analyse the stochastic properties of geodetic time series together with the estimated geophysical signals. The noise spectrum of these time series is generally modelled as a mixed spectrum, with a sum of white and coloured noise. Here, we are interested in modelling the residual time series after deterministically subtracting geophysical signals from the observations. This residual time series is then assumed to be a sum of three stochastic processes, including the family of Lévy processes. The introduction of a third stochastic term models the remaining residual signals and other correlated processes. Via simulations and real time series, we identify three classes of Lévy processes, namely Gaussian, fractional and stable. In the first case, residuals are predominantly constituted of short-memory processes. The fractional Lévy process can be an alternative model to the fBm in the presence of long-term correlations and self-similarity properties. The stable process is here restrained to the special case of infinite variance, which can be only satisfied in the case of heavy-tailed distributions in the application to geodetic time series. Therefore, the model implies potential anxiety in the functional model selection, where missing geophysical information can generate such residual time series.

scholarly journals Application of Lévy Processes in Modelling (Geodetic) Time Series With Mixed Spectra

2020 ◽  
Author(s):  
Jean-Philippe Montillet ◽  
Xiaoxing He ◽  
Kegen Yu ◽  
Changliang Xiong
Keyword(s):  
Time Series ◽  
Lévy Processes ◽  
Stable Process ◽  
Functional Model ◽  
Noise Spectrum ◽  
Levy Processes ◽  
Infinite Variance ◽  
Residual Time ◽  
First Case ◽  
Heavy Tailed

Abstract. Recently, various models have been developed, including the fractional Brownian motion (fBm), to analyse the stochastic properties of geodetic time series, together with the estimated geophysical signals. The noise spectrum of these time series is generally modelled as a mixed spectrum, with a sum of white and coloured noise. Here, we are interested in modelling the residual time series, after deterministically subtracting geophysical signals from the observations. This residual time series is then assumed to be a sum of three stochastic processes, including the family of Lévy processes. The introduction of a third stochastic term models the remaining residual signals and other correlated processes. Via simulations and real time series,we identify three classes of Lévy processes: Gaussian, fractional and stable. In the first case, residuals are predominantly constituted of short-memory processes. Fractional Lévy process can be an alternative model to the fBm in the presence of long-term correlations and self-similarity property. Stable process is here restrained to the special case of infinite variance, which can be only satisfied in the case of heavy-tailed distributions in the application to geodetic time series. Therefore, it implies potential anxiety in the functional model selection where missing geophysical information can generate such residual time series.

On correlating Lévy processes

2010 ◽  
Vol 13 (1) ◽  
pp. 3-16 ◽  
Author(s):  
Ernst Eberlein ◽  
Dilip Madan
Keyword(s):  
Lévy Processes ◽  
Levy Processes

scholarly journals CLASSICAL MARKOV PROCESSES FROM QUANTUM LÉVY PROCESSES

1999 ◽  
Vol 02 (01) ◽  
pp. 105-129 ◽  
Author(s):  
UWE FRANZ
Keyword(s):  
Markov Process ◽  
Hopf Algebra ◽  
Markov Processes ◽  
Lévy Processes ◽  
Sufficient Conditions ◽  
Markov Property ◽  
Vacuum State ◽  
Levy Processes ◽  
Quantum Process ◽  
Quantum Markov Processes

We show how classical Markov processes can be obtained from quantum Lévy processes. It is shown that quantum Lévy processes are quantum Markov processes, and sufficient conditions for restrictions to subalgebras to remain quantum Markov processes are given. A classical Markov process (which has the same time-ordered moments as the quantum process in the vacuum state) exists whenever we can restrict to a commutative subalgebra without losing the quantum Markov property.8 Several examples, including the Azéma martingale, with explicit calculations are presented. In particular, the action of the generator of the classical Markov processes on polynomials or their moments are calculated using Hopf algebra duality.

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