Localizable Moving Average Symmetric Stable and Multistable Processes

We study a particular class of moving average processes that possess a property called localizability. This means that, at any given point, they admit a “tangent process,” in a suitable sense. We give general conditions on the kernel g defining the moving average which ensures that the process is localizable and we characterize the nature of the associated tangent processes. Examples include the reverse Ornstein–Uhlenbeck process and the multistable reverse Ornstein–Uhlenbeck process. In the latter case, the tangent process is, at each time t, a Lévy stable motion with stability index possibly varying with t. We also consider the problem of path synthesis, for which we give both theoretical results and numerical simulations.