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

Abstract

In this paper we consider the stochastic recurrence equation Y_t=A_tY_t−1+B_t for an i.i.d. sequence of pairs (A_t,B_t) of nonnegative random variables, where we assume that B_t is regularly varying with index κ>0 and EA_t^κ<1. We show that the stationary solution (Y_t) to this equation has regularly varying finite-dimensional distributions with index κ. This implies that the partial sums S_n=Y₁+⋯+Y_n of this process are regularly varying. In particular, the relation P(S_n>x)∼c₁nP(Y₁>x) as x→∞ holds for some constant c₁>0. For κ>1, we also study the large deviation probabilities P(S_n−ES_n>x), x≥x_n, for some sequence x_n→∞ whose growth depends on the heaviness of the tail of the distribution of Y₁. We show that the relation P(S_n−ES_n>x)∼c₂nP(Y₁>x) holds uniformly for x≥x_n and some constant c₂>0. Then we apply the large deviation results to derive bounds for the ruin probability ψ(u)=P(sup _n≥1((S_n−ES_n)−μn)>u) for any μ>0. We show that ψ(u)∼c₃uP(Y₁>u)μ⁻¹(κ−1)⁻¹ for some constant c₃>0. In contrast to the case of i.i.d. regularly varying Y_t’s, when the above results hold with c₁=c₂=c₃=1, the constants c₁, c₂ and c₃ are different from 1.