The density of a passage time for a renewal-reward process perturbed by a diffusion

Recently, Coutin and Dorobantu (2011) [1] show the existence of the density for the first passage time τxτx of a level xx by XX, where XX is a Lévy process with a compound Poisson process and a Gaussian component. In this note, we generalize their result and we consider XX a mixed process, the sum of a Brownian motion and a renewal-reward process. Our result (the density of τxτx) may be computed in classical settings (for a Lévy process) and also in a non-Markovian context with possible positive and negative jumps. Compared to Coutin and Dorobantu (2011) [1], we also derive some relations allowing us to build the conditional density when we observe the paths of XX only at jump times. The main advantage of a density formula is that we may obtain the passage time probability with fewer simulations than for an empirical cumulative distribution function of the passage time. Numerical applications illustrate the interest of this result.