On generalized Piterbarg-Berman function
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This paper aims to evaluate the Piterbarg-Berman function given by PB_α^h(x, E) = ∫_Re^zP{∫_E I(√(2)B_α(t) - |t|^α - h(t) - z>0 ) d t > x}d z, x∈[0, mes(E)], with h a drift function and B_α a fractional Brownian motion (fBm) with Hurst index α/2∈(0,1], i.e., a mean zero Gaussian process with continuous sample paths and covariance function Cov(B_α(s), B_α(t)) = 1/2 (|s|^α + |t|^α - |s-t|^α). This note specifies its explicit expression for the fBms with α=1 and 2 when the drift function h(t)=ct^α, c>0 and E=R_+∪{0}. For the Gaussian distribution B_2, we investigate PB_2^h(x, E) with general drift functions h(t) such that h(t)+t^2 being convex or concave, and finite interval E=[a,b]. Typical examples of PB_2^h(x, E) with h(t)=c|t|^λ-t^2 and several bounds of PB_α^h(x, E) are discussed. Numerical studies are carried out to illustrate all the findings. Keywords: Piterbarg-Berman function; sojourn time; fractional Brownian motion; drift function
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