On Bellman's equations for mean and variance control of a Markov diffusion

A controlled diffusion process is considered with cost functions of 'mean and variance' type. A regularization is proposed for computing the value of the cost function via Bellman's equations. The latter equations are in particular useful because they imply sufficiency of markovian strategies, at least, for regularized versions of processes. For the diffusion without control, a system of two well-posed linear partial differential equations (PDEs) is derived, similar to Kac's and Dynkin's moment equations, along with an equivalent single degenerate equation which turns out to be well-posed, too.