Fractional processes conditioned to stay positive
Final Report Abstract
Fractional processes and time series play an important role in modelling dependent data. In particular, fractional Brownian motion is not only a mathematically beautiful object but also a model used in many fields. It is of great interest, again both from a purely mathematical point of view as well as from an applications’ perspective, to study stochastic processes conditioned to stay positive. The difficulty when trying to condition fractional Browian motion or other fractional processes on being positive is that they are intrinsically non-Markovian processes, so that all known techniques (e.g. Doob’s h-transform) fail completely. Instead, we considered a modified problem, where we penalized fractional Brownian motion for being negative – instead of killing it. This is also motivated by a ‘penalizing functional’ that appeared before in the literature on persistence probabilities. The corresponding limit law of the penalized FBM could be obtained. Further, this limiting process could be characterized by a stochastic differential equation in the Brownian case. The relation of the penalized process to a limiting process representing a true fractional Brownian motion conditioned on being positive is yet to be investigated.
Publications
-
Penalizing fractional Brownian motion for being negative. Stochastic Processes and their Applications, 130(11), 6625-6637.
Aurzada, Frank; Buck, Micha & Kilian, Martin
-
Asymptotics of the persistence exponent of integrated fractional Brownian motion and fractionally integrated Brownian motion. Теория вероятностей и ее применения, 67(1), 100-114.
Aurzada, Frank & Kilian, Martin
-
Persistence probabilities of mixed FBM and other mixed processes. Journal of Physics A: Mathematical and Theoretical, 55(30), 305003.
Aurzada, Frank; Kilian, Martin & Sönmez, Ercan
