Predictive Distribution of Anticipative alpha-Stable Markov Processes

The anticipative alpha-stable autoregression of order 1 is a stationary Markov process undergoing explosive episodes akin to bubbles in financial time series data. Although featuring infinite variance, conditional moments up... [ view full abstract ]

The anticipative alpha-stable autoregression of order 1 is a stationary Markov process undergoing explosive episodes akin to bubbles in financial time series data. Although featuring infinite variance, conditional moments up to integer order four may exist. Little is known about their forms and this impedes understanding and forecasting of anticipative processes. The conditional expectation, variance, skewness and kurtosis are provided at any forecast horizon under any admissible parameterisation. During bubble episodes, these moments become equivalent to that of a Bernoulli distribution charging complementary probabilities to two polarly-opposite outcomes: pursued explosion or collapse. The results extend to the continuous time Ornstein-Uhlenbeck.