Diffusion approximation of birth-death processes: Comparison in terms of large deviations and exit points

We compare the large deviations between jump Markov processes and their diffusion approximations for the problem of escape from a domain. It is known that the escape times are generally asymptotically different, leading to a quantitative error. In this paper we show that the escape point can also be different, which implies a qualitative error. In dimension one, for a general class of birth-and-death processes, we show that there always exists a domain [a,b] such that the jump process asymptotically escapes from a and its approximation from b.