  In various practical situations, we encounter data from stochastic processes
which can be efficiently modelled by an appropriate parametric model for
subsequent statistical analyses. Unfortunately, the most common estimation and
inference methods based on the maximum likelihood (ML) principle are
susceptible to minor deviations from assumed model or data contamination due to
their well known lack of robustness. Since the alternative non-parametric
procedures often lose significant efficiency, in this paper, we develop a
robust parameter estimation procedure for discretely observed data from a
parametric stochastic process model which exploits the nice properties of the
popular density power divergence measure in the framework of minimum distance
inference. In particular, here we define the minimum density power divergence
estimators (MDPDE) for the independent increment and the Markov processes. We
establish the asymptotic consistency and distributional results for the
proposed MDPDEs in these dependent stochastic process set-ups and illustrate
their benefits over the usual ML estimator for common examples like Poisson
process, drifted Brownian motion and auto-regressive models.
