  In this paper, we present a discretization algorithm for finite horizon risk
constrained dynamic programming algorithm in [Chow_Pavone_13]. Although in a
theoretical standpoint, Bellman's recursion provides a systematic way to find
optimal value functions and generate optimal history dependent policies, there
is a serious computational issue. Even if the state space and action space of
this constrained stochastic optimal control problem are finite, the spaces of
risk threshold and the feasible risk update are closed bounded subset of real
numbers. This prohibits any direct applications of unconstrained finite state
iterative methods in dynamic programming found in [Bertsekas_05]. In order to
approximate Bellman's operator derived in [Chow_Pavone_13], we discretize the
continuous action spaces and formulate a finite space approximation for the
exact dynamic programming algorithm. We will also prove that the approximation
error bound of optimal value functions is bound linearly by the step size of
discretization. Finally, details for implementations and possible modifications
are discussed.
