The maximum surplus before ruin in a jump-diffusion insurance risk process with dependence

Abstract

We consider a compound Poisson risk process perturbed by a Brownian motion through using a potential measure where the claim sizes depend on inter-claim times via the Farlie-Gumbel-Morgenstern copula. We derive an integro-differential equation with certain boundary conditions for the distribution of the maximum surplus before ruin. This distribution can be calculated through the probability that the surplus process attains a given level from the initial surplus without first falling below zero. The explicit expressions for this distribution are derived when the claim amounts are exponentially distributed.

Valuation of Cliquet-Style Guarantees with Death Benefits in Jump Diffusion Models

“…An alternative is to use exponential Lévy models. Extra jumps could be added to the drifted Brownian motion in the other areas of financial mathematics, see [9][10][11] for example. In particular, when the distribution of jumps follows a combination (or a mixture) of exponential distributions, the jump diffusion (2) becomes highly tractable, see [12][13][14], therein.…”