Detecting Extreme Events in Lévy processes

In our paper we develop a length scale LbL_{\text{b}} that we call the "bifurcation length". Below is a plot of this LbL_{\text{b}} in the natural units σT1/α\sigma T^{1/\alpha} of a Levy process, where σ\sigma is the width parameter, α\alpha is the stability index, and TT is the evolution time.

Plot of levy process bifurcation length.

Notice that LbL_{\text{b}} diverges as α2\alpha\rightarrow 2, demonstrating that this is indeed a scale that is distinct from the natural length scale.

To see the effectiveness of this scale for detecting extreme events, below I show a plot of a Levy process sample path (Δt=T/500\Delta t=T/500 with α=1.9\alpha=1.9) with certain "extreme" steps highlighted. You can highlight steps based on the bifurcation-length criteria (Δx>Lb(Δt/T)1/α\Delta x > L_{\text{b}}(\Delta t/T)^{1/\alpha}) or a more naive criteria (Δx>σ(Δt/T)1/α\Delta x > \sigma(\Delta t/T)^{1/\alpha}) by hovering your mouse over the corresponding indicator.

Plot of levy flight with a=1.9

Qualitatively, we see that steps that appear "extreme" are well indicated by the bifurcation-length based criteria, while the naive criteria selects many steps that do not appear to be extreme by eye.

This also work for other values of α\alpha, (α=1.5\alpha=1.5 shown below), though as α\alpha approaches 11 both the bifurcation-length based criteria and the naive criteria indicate the same steps as "extreme".

Plot of levy flight with a=1.5

For the derivation and justification of this scale, see our paper here.