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.