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

For a general penalty function, the expected discounted penalty at ruin was considered by, for example, Gerber and Shiu(1998) and Gerber and Landry (1998) in insurance literature. On the other hand, many pricing functionals in mathematical finance(e.g., options pricing, credit risk modelling) can be formulated in terms of expected discounted penalties. Under the assumption that the asset value follows a jump diffusion, we show the expected discounted penalty satisfies a homogeneous ODE. Based on ODE theory, we obtain a general form for the expected discounted penalty. In particular, if only downward phase-type jumps are allowed, we obtain an explicit formula in terms of the penalty function. On the other hand, if downward jump distribution is a mixture of exponential distributions (and upward jumps are determined by a general Lévy measure), we obtain closed form solutions for the expected discounted penalty. For earlier and related results, see Gerber and Landry(1998), Hilberink and Rogers(2002), Mordecki(2002), Kou and Wang(2004), Asmussen et al.(2004) and others.