Geodesic
On control of a diffusion process up to the time of first exit from a region
Izvestiya. Mathematics, Tome 19 (1982) no. 2, pp. 297-313 
N. V. Krylov. On control of a diffusion process up to the time of first exit from a region. Izvestiya. Mathematics, Tome 19 (1982) no. 2, pp. 297-313. http://geodesic.mathdoc.fr/item/IM2_1982_19_2_a5/
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Voir la notice de l'article provenant de la source Math-Net.Ru

The problem of controlling a diffusion process with smooth coefficients up to the time of first exit from a region is reduced to the problem of controlling a diffusion process with smooth coefficients in a whole space. Three examples are considered which show the usefulness of such a reduction in deriving the Bellman equation for the original problem. Bibliography: 12 titles.

[1] Safonov M. V., “On the control of diffusion processes in a multidimensional cylindrical domain”, Intern. Symp. Stoch. Diff. Eq., Abstr. Comm. (1978, Vilnius), In-t matem. i kibern. Litovsk. AN, 1978, 168–172

[2] Safonov M. V., “O zadache Dirikhle dlya uravneniya Bellmana v mnogomernoi oblasti”, Dokl. AN SSSR, 253:3 (1980), 535–640 | MR

[3] Lions P.-L., “Equations de Hamilton–Jacobi–Bellman dégénerées”, Comptes Rendus Acad. Sc., série A, 289 (1979), 329–332 | MR | Zbl

[4] Evans L. C, Friedman A., “Optimal stochastic switching and the Dirichlet problem for the Bellman equations”, Trans. Amer. Math. Soc., 253 (1979), 365–389 | DOI | MR | Zbl

[5] Evans L. C., Second derivative estimates for the Bellman equation, Preprint

[6] Brezis H., Evans L. C., “A variational inequality approach to the Bellman–Dirichlet equation for two elliptic operators”, Archive Rat. Mech. and Anal., 71:1 (1979), 1–13 | DOI | MR | Zbl

[7] Krylov N. V., Upravlyaemye protsessy diffuzionnogo tipa, Nauka, M., 1977 | MR

[8] Krylov N. V., “Ob upravlyaemykh diffuzionnykh protsessakh s neogranichennymi koeffitsientami”, Izv. AN SSSR. Ser. matem., 45:4 (1981), 734–759 | MR | Zbl

[9] Krylov N. V., “Control of the diffusion type processes”, Proc. Internat. Congress Math., v. 2 (Helsinki, 1978), 1980, 859–863 | MR | Zbl

[10] Krylov N. V., “Nekotorye novye rezultaty iz teorii upravlyaemykh diffuzionnykh protsessov”, Matem. sb., 109:1 (1979), 146–164 | MR | Zbl

[11] Krylov N. V., “Obschie metody otsenki proizvodnykh funktsii vyigrysha”, Veroyatnostnye protsessy i upravlenie, Nauka, M., 1978, 94–125 | MR

[12] Cheng Shiu-Yuen, Yau Shing-Tung, “On the regularity of the Monge–Ampère equation $\mathrm{det}(\partial^2u/\partial x_i\partial x_j)=F(x,u)$”, Comm. Pure and Appl. Math., 30:1 (1977), 41–68 | DOI | MR | Zbl