  Different change-point type models encountered in statistical inference for
stochastic processes give rise to different limiting likelihood ratio
processes. In this paper we consider two such likelihood ratios. The first one
is an exponential functional of a two-sided Poisson process driven by some
parameter, while the second one is an exponential functional of a two-sided
Brownian motion. We establish that for sufficiently small values of the
parameter, the Poisson type likelihood ratio can be approximated by the
Brownian type one. As a consequence, several statistically interesting
quantities (such as limiting variances of different estimators) related to the
first likelihood ratio can also be approximated by those related to the second
one. Finally, we discuss the asymptotics of the large values of the parameter
and illustrate the results by numerical simulations.
