Omega Model for a Jump-Diffusion Process with a Two-Step Premium Rate and a Threshold Dividend Strategy

In this paper, a jump-diffusion Omega model with a two-step premium rate and a threshold dividend strategy is studied. For this model, the surplus process is a perturbation of a compound Poisson process by a Brownian motion. Firstly, using the strong Markov property, the integro-differential equations for the expected discounted dividend payments function, the Gerber-Shiu expected discounted penalty function and bankruptcy probability are derived. Secondly, for a constant bankruptcy rate function, the renewal equations satisfied by the expected discounted dividend payments function and the Gerber-Shiu expected discounted penalty function are obtained, respectively, and by iteration, their closed-form solutions are also given. Furthermore, the explicit solutions of the two kinds of functions are obtained when the individual claim size is subject to exponential distribution. Finally, a numerical example is presented to illustrate some properties of the Omega model.