Retrodicting with the truncated Lévy flight

Abstract

There is a point in predicting the past (retrodicting) because we lack information about it. To address this issue, we consider a truncated Lévy flight to model data. We build on the finding that there is a power law between truncation length and standard deviation that connects the bounded past and unbounded future. Even if a truncated Lévy flight cannot predict future extreme events, we argue that it can still be used to model the past. Because we avoid the exact form of the probability density function while allowing its distributional moments, with the exception of the mean, to vary over time, our method is applicable to a wide range of symmetric distributions. We illustrate our point by using US dollar prices in 15 different currencies traded on foreign exchange markets.