Geodesic
Some new results in the theory of controlled diffusion processes
Sbornik. Mathematics, Tome 37 (1980) no. 1, pp. 133-149 
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/
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Voir la notice de l'article provenant de la source Math-Net.Ru

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.

[1] N. V. Krylov, Upravlyaemye protsessy diffuzionnogo tipa, izd-vo “Nauka”, Moskva, 1977 | MR

[2] W. H. Fleming, “The Cauchy problem for degenerate parabolic equations”, J. Math. Mech., 13:6 (1964), 987–1008 | MR | Zbl

[3] W. H. Fleming, “The Cauchy problem for a nonlinear first-order partial differential equation”, J. Diff. Equat., 5:3 (1969), 515–530 | DOI | MR | Zbl

[4] S. N. Kruzhkov, Nelineinye uravneniya s chastnymi proizvodnymi, ch. 2, izd. MGU, Moskva, 1970

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[6] A. I. Volpert, S. I. Khudyaev, “O zadache Koshi dlya kvazilineinykh vyrozhdayuschikhsya parabolicheskikh uravnenii vtorogo poryadka”, Matem. sb., 78(120) (1969), 374–396 | MR

[7] N. V. Krylov, “O predelnom perekhode v vyrozhdennykh uravneniyakh Bellmana, I”, Matem. sb., 106(148) (1978), 214–233 | MR | Zbl

[8] N. V. Krylov, “O predelnom perekhode v vyrozhdennykh uravneniyakh Bellmana. II”, Matem. sb., 107(149) (1978), 56–68 | MR | Zbl

[9] O. A. Oleinik, “Razryvnye resheniya nelineinykh differentsialnykh uravnenii”, Uspekhi matem. nauk, XII:3(75) (1957), 3–73 | MR

[10] E. Conway, E. Hopf, “Hamilton's theory and generalized solutions of the Hamilton–Jacoby equation”, J. Math. Mech., 13:6 (1964), 939–986 | MR | Zbl

[11] O. A. Oleinik, E. V. Radkevich, “Uravneniya vtorogo poryadka s neotritsatelnoi kharakteristicheskoi formoi”, Itogi nauki. Matematicheskii analiz 1969, 8, VINITI, Moskva, 1971, 7–252 | MR | Zbl

[12] A. D. Aleksandrov, “Suschestvovanie pochti vezde vtorogo differentsiala vypukloi funktsii i nekotorye svyazannye s nim svoistva vypuklykh poverkhnostei”, Uchenye zapiski LGU, 37, seriya matem., 1939, no. 6, 3–35 | Zbl

[13] N. V. Krylov, “Nekotorye svoistva normalnogo izobrazheniya vypuklykh funktsii”, Matem. sb., 105(147) (1978), 180–191 | MR | Zbl