  Continuous-discrete models with dynamics described by stochastic differential
equations are used in a wide variety of applications. For these systems, the
maximum a posteriori (MAP) state path can be defined as the curves around which
lie the infinitesimal tubes with greatest posterior probability, which can be
found by maximizing a merit function built upon the Onsager--Machlup
functional. A common approach used in the engineering literature to obtain the
MAP state path is to discretize the dynamics and obtain the MAP state path for
the discretized system. In this paper, we prove that if the trapezoidal scheme
is used for discretization, then the discretized MAP state path estimation
converges hypographically to the continuous-discrete MAP state path estimation
as the discretization gets finer. However, if the stochastic Euler scheme is
used instead, then the discretized estimation converges to the minimum energy
estimation. The minimum energy estimates are, in turn, proved to be the state
paths associated with the MAP noise paths, which in some cases differ from the
MAP state paths. Therefore, the discretized MAP state paths can have different
interpretations depending on the discretization scheme used.
