Persistence probabilities for fractionally integrated fractional Brownian noise

The main objective of this study is fractionally integrated fractional Brownian noise, I(t/a,H) where a>0 is the 'multiplicity' of integration, and H is the Hurst parameter . The subject of the analysis is the persistence exponent e(a,H) that determines the power-law asymptotic of probability that the process will not exceed a fixit level in a growing time interval (0,T). In the important cases such as fractional Brownian motion(a=1,H) and integtated Wienr process(a=2,H=1/2) these exponents are well known. To understand the problematic exponents e(2,H), we consider the (a,H) parameters from the maximum (for the task) area G= (a+H>1,0<H<1) ). We prove the decrease of the exponents with increasing 'a' and describe their behavior near the boundary of G, including infinity. The discovered identity of the exponents with the parameters (a,H) and (a+2H-1,1-H) actually refutes the long-standing hypothesis that e(2,H)=H(1-H). Our results use well known the continuity lemma for the persistence exponents and a some generalization of Slepian's lemma for a family of Gaussian processes smoothly dependent on a parameter.