An ODE approach for the expected discounted penalty at ruin in a jump-diffusion model

Abstract

Under the assumption that the asset value follows a phase-type jump-diffusion, we show that the expected discounted penalty satisfies an ODE and obtain a general form for the expected discounted penalty. In particular, if only downward jumps are allowed, we get an explicit formula in terms of the penalty function and jump distribution. On the other band, if the downward jump distribution is a mixture of exponential distributions (and upward jumps are determined by a general Levy measure), we obtain closed-form solutions for the expected discounted penalty. As an application, we work out an example in Leland's structural model with jumps. For earlier and related results, see Gerber and Landry [Insur. Math. Econ. 22:263276, 1998], Hilberink and Rogers [Finance Stoch. 6:237-263, 2002], Asmussen et al. [Stoch. Proc. Appl. 109:79-111, 2004], and Kyprianou and Surya [Finance Stoch. 11:131-152, 2007].