The Kalman-Lévy filter (Q5942848)

Conclusion of the authors: We have presented the solution of the Kalman filter problem for dynamical and forecast noises distributed according to power laws and Lévy laws. The main theoretical concept that we have used is to optimize the Kalman filter to chisel the tail of the distribution of residual errors so as to minimize it globally. In order to implement this program, we have introduced the concept of a ``tail-covariance'' that generalizes the usual notion of covariance. The full solution, called the KL filter, is obtained by the solution of a general nonlinear equation. We have investigated in detail the quality of this solution in the univariate case and have shown by direct numerical experiments that the improvement is significant, all the more so, the heavier the tail, i.e. the smaller the power law exponent \(\mu\).