Pickands-Piterbarg constants for self-similar Gaussian processes

For a centered self-similar Gaussian process $\{Y(t):t\in [0,\mathrm{\infty })\}$ and $R\ge 0$ we analyze asymptotic behaviour of $${\mathcal{H}}_Y^R(T)=\mathbf{E}\exp (\underset{t\in [0,T]}{\sup }\sqrt{2}Y(t)-(1+R){\sigma }_Y^2(t)),$$ as $T\to \mathrm{\infty }$. We prove that $$\mathcal{H}$Y^R=\lim{T\to\infty} $\mathcal{H}$_Y^R(T)\in(0,\infty)$for$R>0$ and $${\mathcal{H}}_Y=\underset{T\to \mathrm{\infty }}{\lim }\frac{{\mathcal{H}}_Y^0(T)}{T^{\gamma }}\in (0,\mathrm{\infty })$$ for suitably chosen $\gamma >0$. Additionally, we find bounds for ${\mathcal{H}}_Y^R$, $R>0$ and a surprising relation between ${\mathcal{H}}_Y$ and classical Pickands constants.