Escape probabilities for non-Markovian point processes, a panoramic view

J. Villarroel, J. A. Vega Coso, M. Montero

Feller [1] showed how to determine escape probabilities from intervals for continuous time diffusion processes. Unfortunately, no such well established theory exists for non-Markovian point processes. Here we give a panoramic view of existing results and partial theories (Sparre, Bertoin, Oksendal, Dickson) and show how to solve this problem in the case of compound renewal processes in continuous time ($X_t$) whose dynamics combines uniform motion with speed c and sudden jumps $J_n$ at time points $t_n$ triggered by a renewal process ($N_t$, t>0). Here $N_t$ counts the number of events $t_n$, n=1,..infinity "observed'' in the time window (0,t] and $X_t= ct + J_1+..J_n if N_t=n$. We formulate integral equations that satisfy escape probabilities. In the case of purely negative jumps we give closed form expressions for this probability.
1. W. Feller, Diffusion processes in one dimension. Trans. Amer. Math. Soc., (77), 1-3 (1954)

Keywords: Escape and ruin probabilities renewal reward process

J. Villarroel Rodríguez, J. A. Vega Coso, J. A. Vega Coso

A. Gómez Corral

A. Gómez Corral