Persistence probabilities for fractionally integrated fractional Brownian noise

Abstract

The main objective of this study is fractionally integrated fractional Brownian noise, I(t/a,H) where a>0 is the &#39;multiplicity&#39; 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(FBM(H),a=1) 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 &#39;a&#39; and describe their behavior near the boundary of G, including infinity. The identity of the exponents with parameters (a,H) and (a+2H-1,1-H) has been established. On this way, the long-standing hypothesis that e(2,H)=H(1-H) has been refuted. In addition, it has been revealed that FBM(H) and FBM(1-H) processes are related by a fractional integration operation. We have obtained the exact value of the exponent for e(a>>1, H). It is identical to the persistence exponent for a Gaussian stationary process with covariance cosh((H-1/2)t)/cosh(t/2) and generalizes the well-known case of H=1/2. Our results use well known the continuity lemma for the persistence exponents and a some generalization of Slepian&#39;s lemma for a family of Gaussian processes smoothly dependent on a parameter.