On the ruin probability for physical fractional Brownian motion

On the infimum attained by the reflected fractional Brownian motion

“…Remark 1 Conditions similar to assumption (8) have been introduced in, among others, Dȩbicki (2002), Hüsler and Piterbarg (2004), Dȩbicki and Kisowski (2008), and Dȩbicki and Tabiś (2011) as a standard way of capturing nonstationarity. The shape of Lemma 1 is tailored to the needs of the next section, where asymptotics of tail distribution of inf sup functionals of Gaussian processes are analyzed.…”

Extreme values of a portfolio of Gaussian processes and a trend

“…The ruin occurs if at some time t the sum of the claims is larger than the sum of premium payments and the reserve. This problem is investigated for a class of Gaussian processes XðtÞ (including fractional Brownian motion and self-similar Gaussian processes) in Hü sler and Piterbarg (1999) and for integrated Gaussian processes in Debicki (2002) as well as Hü sler and Piterbarg (2004). In these cases the probability PðuÞ is approximated by exceedances of YðtÞ in a small neighborhood of a unique point where the boundary ðu þ ct Þ='ðtÞ has smallest value.…”

On average losses in the ruin problem with fractional Brownian motion as input

“…The ruin is said to occur on the interval [0, T] if inf t∈[0,T] S t < 0. The probability of ruin is investigated in more general Gaussian setting in Hüsler and Piterbarg (1999), and for integrated Gaussian processes in Dieker (2005), Kobelkov (2004), Hüsler and Piterbarg (2004). In Hüsler and Piterbarg (2008) a limit result for the time of ruin is obtained.…”