On modeling quantities for insurer solvency against catastrophe under some Markovian assumptions

Insurance companies sometimes face catastrophic losses, yet they must remain solvent enough to meet the legal obligation of covering all claims. Catastrophes can result in large damages to the policyholders, causing the arrival of numerous claims to insurance companies at once. Furthermore, the severity of an event could impact the time until the next occurrence. An insurer needs certain levels of startup capital to meet all claims, and then must have adequate reserves on a continual basis, even more so when catastrophes occur. This work examines two facets of these matters: for an infinite time horizon, we extend and develop models for insurer bankruptcy-related quantities accounting for the reality of large claims occurring. Meanwhile, for finite time horizons, we model the present value of claims that have been incurred but not yet reported, so-called \u27IBNR\u27 claims. In the former, we show how our method for \u27Gerber-Shiu\u27 functions works in a recently proposed dependency structure allowing insurers to charge clients different premiums depending on their riskiness. In the latter, we build upon a recent method which allowed claims to arrive in batches; besides permitting discounting to be time-dependent, we allow the insurer to adjust the assumed distribution of the time until the next event by comparing the number of claims from the current event to any number of random intervals. We provide numerical studies for both scenarios --Abstract, page iii