The Kalman–Lévy filter

The Kalman filter combines forecasts and new observations to obtain an estimation which is optimal in the sense of a minimum average quadratic error. The Kalman filter has two main restrictions: (i) the dynamical system is assumed linear and (ii) forecasting errors and observational noises are projected onto Gaussian distributions. Here, we offer an important generalization to the case where errors and noises have heavy tail distributions such as power laws and Lévy laws. The main tool needed to solve this “Kalman–Lévy” filter is the “tail-covariance” matrix which generalizes the covariance matrix in the case where it is mathematically ill-defined (i.e. for power law tail exponents μ≤2). We present the general solution and discuss its properties on pedagogical examples. The standard Kalman–Gaussian filter is recovered for the case μ=2. The optimal Kalman–Lévy filter is found to deviate substantially from the standard Kalman–Gaussian filter as μ deviates from 2. As μ decreases, the filter favors more strongly the better one of the forecast and the observation, based on the tail-covariance matrix because a small exponent μ implies large errors with significant probabilities. In terms of implementation, the price-to-pay associated with the presence of heavy tail noise distributions is that the standard linear formalism valid for the Gaussian case is transformed into a nonlinear matrix equation for the Kalman–Lévy filter. Direct numerical experiments in the univariate case confirms our theoretical predictions.