Geodesic
Some new results in the theory of controlled diffusion processes
Sbornik. Mathematics, Tome 37 (1980) no. 1, pp. 133-149

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The validity of the Bellman equation for the payoff function for a controlled random diffusion process is proved. The main difference between the results in this article and those known earlier on the same theme is that here no assumptions whatever are made on the nondegeneracy of the controlled process. A theorem on the uniqueness of the solution of the Bellman equation is proved as well. The Bellman equation is examined in a lattice of measures; the derivatives of functions on which it is studied are understood as measures. Bibliography: 13 titles. 
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     title = {Some new results in the theory of controlled diffusion processes},
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N. V. Krylov. Some new results in the theory of controlled diffusion processes. Sbornik. Mathematics, Tome 37 (1980) no. 1, pp. 133-149. http://geodesic.mathdoc.fr/item/SM_1980_37_1_a8/