  This paper considers robust Markov decision processes under parametric
transition distributions. We assume that the true transition distribution is
uniquely specified by some parametric distribution, and explicitly enforce that
the worst-case distribution from the model is uniquely specified by a
distribution in the same parametric family. After formulating the parametric
robust model, we focus on developing algorithms for carrying out the robust
Bellman updates required to complete robust value iteration. We first formulate
the update as a linear program by discretising the ambiguity set. Since this
model scales poorly with problem size and requires large amounts of
pre-computation, we develop two additional algorithms for solving the robust
Bellman update. Firstly, we present a cutting surface algorithm for solving
this linear program in a shorter time. This algorithm requires the same
pre-computation, but only ever solves the linear program over small subsets of
the ambiguity set. Secondly, we present a novel projection-based bisection
search algorithm that completely eliminates the need for discretisation and
does not require any pre-computation. We test our algorithms extensively on a
dynamic multi-period newsvendor problem under binomial and Poisson demands. In
addition, we compare our methods with the non-parametric phi-divergence based
methods from the literature. We show that our projection-based algorithm
completes robust value iteration significantly faster than our other two
parametric algorithms, and also faster than its non-parametric equivalent.
