  A variant of the optimal control problem is considered which is nonstandard
in that the performance index contains "stochastic" integrals, that is,
integrals against very irregular functions. The motivation for considering such
performance indices comes from dynamical estimation problems where observed
time series need to be "fitted" with trajectories of dynamical models. The
observations may be contaminated with white noise, which gives rise to the
nonstandard performance indices. Problems of this kind appear in engineering,
physics, and the geosciences where this is referred to as data assimilation.
Pathwise existence of minimisers is obtained, along with a maximum principle as
well as preliminary results in dynamic programming. The results extend previous
results on the maximum aposteriori estimator of trajectories of diffusion
processes. To obtain these results, classical concepts from optimal control
need to be substantially modified due to the nonstandard nature of the
performance index, as well the fact that typical models in the geosciences do
not satisfy linear growth nor monotonicity conditions.
