Finite-Time Ruin Probabilities via Multivariate Lagrangian Inversion

In this paper, we study finite-time ruin probabilities for the compound Markov binomial risk model - a discrete model with claim sizes depending on a finite-state ergodic Markov chain. We show that a general form of the Ballot Theorem remains valid under stationarity of the modulating chain, yielding a Takacs-type expression for the finite-time ruin probability which holds only when the initial surplus is equal to zero. For the general, non-stationary case with arbitrary initial surplus, the Ballot Theorem can no longer be employed and thus, we develop an approach based on multivariate Lagrangian inversion to derive distributional results for various hitting times of the risk process, including Seal-type and Picard-Lefevre-type formulas for the finite-time ruin probability.