On corrected phase-type approximations of the time value of ruin with heavy tails

We approximate Gerber-Shiu functions with heavy-tailed claims in a recently introduced risk model having both interclaim times and premiums depending on the claim sizes. We apply a technique known as "corrected phase-type approximations". This results in adding a correction term to the Gerber-Shiu function with phase-type claim sizes. The correction term contains the heavy-tailed behavior at most once per convolution and captures the tail behavior of the true Gerber-Shiu function. We make the tail behavior specific in the classical case of one class of risk insured. After illustrating a use of such approximations, we study numerically the approximations' relative errors for some specific penalty functions and claims distributions.