Inverse Problems under Sarmanov dependence structure

Abstract

Consider a sequence $\{(X_{i}, Y_{i})\}$ of independent and identically distributed random vectors, with joint distribution bivariate Sarmanov. This is a natural set-up for discrete time financial risk models with insurance risks. Of particular interest are the infinite time ruin probabilities $P\left[\sup_{n \geq 1}\sum_{i=1}^{n} X_i \prod_{j=1}^{i}Y_{j} > x\right]$. When the $Y_{i}$'s are assumed to have lighter tails than the $X_{i}$'s, we investigate sufficient conditions that ensure each $X_{i}$ has a regularly varying tail, given that the ruin probability is regularly varying. This is an inverse problem to the more traditional analysis of the ruin probabilities based on the tails of the $X_{i}$'s. We impose moment-conditions as well as non-vanishing Mellin transform assumptions on the $Y_{i}$'s in order to achieve the desired results. But our analysis departs from the more conventional assumption of independence between the sequences $\{X_{i}\}$ and $\{Y_{i}\}$, instead assuming each $(X_{i}, Y_{i})$ to be jointly distributed as bivariate Sarmanov, a fairly broad class of bivariate distributions.