Large deviations of infinite intersections of events in Gaussian processes

Consider events of the form { Z s ≥ ζ ( s ) , s ∈ S } , where Z is a continuous Gaussian process with stationary increments, ζ is a function that belongs to the reproducing kernel Hilbert space R of process Z , and S ⊂ R is compact. The main problem considered in this paper is identifying the function β ∗ ∈ R satisfying β ∗ ( s ) ≥ ζ ( s ) on S and having minimal R -norm. The smoothness (mean square differentiability) of Z turns out to have a crucial impact on the structure of the solution. As examples, we obtain the explicit solutions when ζ ( s ) = s for s ∈ [ 0 , 1 ] and Z is either a fractional Brownian motion or an integrated Ornstein–Uhlenbeck process.