Large deviations and ruin probabilities for solutions to stochastic recurrence equations with heavy-tailed innovations

摘要

In this paper we consider the stochastic recurrence equation Y-t = A(t)Y(t-1) + B-t for an i.i.d. sequence of pairs (A(t), B-t) of nonnegative random variables, where we assume that Bt is regularly varying with index kappa > 0 and EA(t)(kappa) < 1. We show that the stationary solution (Y-t) to this equation has regularly varying finite-dimensional distributions with index K. This implies that the partial sums S-n = Y-1 + - - - + Y-n of this process are regularly varying. In particular, the relation P(S-n > x) similar to c(1)nP(Y-1 > x) as x -> infinity holds for some constant c(1) > 0. For kappa > 1, we also study the large deviation probabilities P(S-n - ESn > x), x >= x(n), for some sequence x(n) -> infinity whose growth depends on the heaviness of the tail of the distribution of Y-1. We show that the relation P(S-n - ESn > x) similar to c(2)nP(Y-1 > x) holds uniformly for x >= x(n) and some constant c(2) > 0. Then we apply the large deviation results to derive bounds for the ruin probability psi(u) = P(sup(n >= 1) ((S-n - ESn) - mu n) > u) for any mu > 0. We show that psi(u) similar to c(3)uP(Y-1 > u)mu(-1) (kappa - 1)(-1) for some constant c(3) > 0. In contrast to the case of i.i.d. regularly varying Y-t's, when the above results hold with c(1) = c(2) = c(3) = 1, the constants c(1), c(2) and c(3) are different from 1.