Integrals of Bessel processes and multi-dimensional Ornstein-Uhlenbeck processes: exact asymptotics for L-p-functionals

摘要

We prove results on exact asymptotics of the expectations E-a exp(-integral(T)(0) xi(p()(q)t) dt) vertical bar xi(q)(T) = b] as T - infinity for p 0, a = 0, b = 0, where xi(q)(t), t = 0, is a Bessel process of order q = -1/2. We also find exact asymptotics of the probabilities P{integral(1)(0) Sigma(n)(k=1) vertical bar Yk(t)vertical bar(p) dt = epsilon(p)}, P{integral(1)(0)[Sigma(n)(k=1) Y-k(2)(t)(p/2) dt = epsilon(p)} as epsilon - 0, where Y(t) - (Y-1(t),..., Y-n(t)), t = 0, is the n-dimensional non-stationary Ornstein-Uhlenbeck process with a parameter gamma = (gamma(1),..., gamma(n)) starting at the origin. We also obtain a number of other results. Numerical values of the asymptotics are given for p = 1, p = 2.