RPA pathwise derivative estimation of ruin probabilities
The surplus process of an insurance portfolio, defined as the wealth obtained by the premium payments minus the reimbursements made at the times of claims, is studied in the case where the wealth available is invested at a continuously compounded rate \(\delta\): \[ dU(t) = (c + \delta U(t)) dt - dS(t); \quad U(0) = u. \] Here \(U(t) = u + ct + S(t)\) and \(S(t) = \sum_{i=1}^{N(t)} {Y_i}\) where \(N(t)\) is a Poisson process with rate \(\lambda\) that models the epochs when claims are made and \(Y_i\) (the corresponding amounts) are i.i.d. random variables; premiums being received at constant rate \(c\) [cf., e.g., \textit{H. Gerber}, An introduction to mathematical risk theory. Monograph 8, Univ. Pennsylvania, Philadelphia (1979; Zbl 0431.62066)]. In general, for the ruin probability: \[ \psi(u, \lambda) = P\{ \text{min} \{t: U(t) < 0\} < \infty \} \] there is no analytical expression available. The paper under review focuses on the estimation of the sensivities of \(\psi(u, \lambda)\) to the arrival rate \(\lambda\) via the rare perturbation analysis method applied to related: regenerative storage process as well as to importance sampling. For phantom and virtual estimators recursion formulas are provided to program these methods efficiently. There are also found computer simulation results for the exponential claim distribution, in order to compare the two estimation methods presented.
scientific article; zbMATH DE number 1525147
scientific article