Persistence probabilities of mixed FBM and other mixed processes

Abstract

We consider the sum of two self-similar centred Gaussian processes with different self-similarity indices. Under the assumption of non-negative correlations and some further minor conditions, we show that the asymptotic behaviour of the persistence probability of the sum is the same as for the process with the greater self-similarity index. In particular, this covers the mixed fractional Brownian motion introduced in (Cheridito 2001 Bernoulli 7 913–34) and shows that the corresponding persistence probability decays asymptotically polynomially with persistence exponent 1 − max(1/2, H), where H is the Hurst parameter of the underlying fractional Brownian motion.