This paper considers the transformation of the digital optimal control problem to an equivalent discrete-time optimal control problem for linear continuous-time systems with continuous-time white stochastic parameters and additive continuous-time white noise. The observations available at the sampling instants are in general non-linear and corrupted by discrete-time noise. The equivalent discrete-time system has white stochastic parameters. Expressions are derived for the first and second moments of these parameters and for the cost criterion parameters which are explicit in the parameters and the statistics of the continuous-time system.