PREDICTION FORMULAS FOR INTEGRATED FRACTIONAL BROWNIAN MOTION

This paper investigates the projection (prediction) of increments of the integrated fractional Brownian motion (ifBm). We introduce ifBm, calculate its covariance function, and establish the stationarity of its increments. Our primary goal is to determine the coefficients for the linear prediction of a future ifBm increment based on a series of past increments. We show that the prediction coefficients exhibit complex behavior, with their signs changing depending on the Hurst index (H). For overlapping increments, this sign change occurs at a non-trivial values of H. In contrast, for non-overlapping increments, the sign change happens at H=0.5, consistent with the properties of fractional Brownian motion itself. This work provides explicit formulas where possible and extensive numerical tables, offering insights into the properties and predictability of ifBm, laying the groundwork for further research.