Taylor-series expansion for multivariate characteristics of classical risk processes

For the continuous-time risk model with compound Poisson input, the (finite-horizont) joint probability P(τ ≤ τ, X ≤ x, Y ≤ y) of ruin time τ, surplus X just before ruin and deficit Y at ruin time τ is considered as a function of the arrival rate λ of claims. It is expanded into a Taylor series at λ = 0. A certain extension of a corresponding result for infinite-horizont joint probabilities, which previously has been derived in Gerber et al. (1987), is also given. For each n ≥ 1, the coefficient of λn is determined by using a general representation formula for the derivatives of a wide class of functionals of independently marked Poisson processes.