On conditional distributions of diffusion processes

N. V. Krylov ; B. L. Rozovskii

Izvestiya. Mathematics , Tome 12 (1978) no. 2, pp. 336-356.

Résumé

For a two-component diffusion process $(x,y)$ on the Euclidean space $R^n$ ($n\geqslant2$), we consider the question of the existence of the density $\pi_{t,s}$ of the distribution $P(x_t\in\nobreak\cdot\,|\,y_\tau,\ \tau\leqslant s)$, $s\leqslant t$, with respect to Lebesgue measure, and we study its analytic properties. We also consider the question of the existence and uniqueness of the solution of the equation for $\pi_{t,t}$ (the filtering equation). Bibliography: 18 titles.

Bibliographie

[1] Fleming W. H., “Optimal Control of Partially Observable Diffusions”, SIAM J. Control, 1968, no. 2, 194–214 | DOI | MR | Zbl

[2] Wonham W. M., “On the Separation Theorem of Stochastic Control”, SIAM J. Control, 1968, no. 2, 312–326 | DOI | MR | Zbl

[3] Liptser R. Sh., Shiryaev A. N., Statistika sluchainykh protsessov, Nauka, M., 1974 | MR | Zbl

[4] Liptser R. Sh., “Optimalnoe kodirovanie i dekodirovanie pri peredache gaussovskogo markovskogo signala po kanalu s besshumnoi obratnoi svyazyu”, Problemy peredachi informatsii, 10:4 (1974), 3–15 | Zbl

[5] Struk D. V., Varadan S. R., “Diffuzionnye protsessy s nepreryvnymi koeffitsientami. II”, Matematika, 16:2 (1972), 100–142

[6] Krylov N. V., “Nekotorye otsenki plotnosti raspredeleniya stokhasticheskogo integrala”, Izv. AN SSSR. Ser. matem., 38 (1974), 228–248 | MR | Zbl

[7] Rozovskii B. L., “O stokhasticheskikh differentsialnykh uravneniyakh v chastnykh proizvodnykh”, Matem. sb., 96(138):2 (1975), 314–341 | MR | Zbl

[8] Levieux F., Recursive Non-linear Filtering: A Solution of the Mathematical Problem and Applications, preprints of IFAC Symp. on Stochastic Control, Budapest, 1974, 175–182

[9] Krylov N. V., Rozovskii B. L., “O zadache Koshi dlya lineinykh stokhasticheskikh differentsialnykh uravnenii s chastnymi proizvodnymi”, Izv. AN SSSR. Ser. matem., 41 (1977), 1329–1347 | MR | Zbl

[10] Iosida K., Funktsionalnyi analiz, Mir, M., 1967 | MR

[11] Struk D. V., Varadan S. R., “Diffuzionnye protsessy s nepreryvnymi koeffitsientami. I”, Matematika, 15:6 (1971), 66–113

[12] Fujisaki M., Kallianpur G., Kunita H., “Stochastic Differential Equations for the Nonlinear Filtering Problem”, Osaka J. Math., 9:1 (1972), 19–40 | MR | Zbl

[13] Rozovskii B. L., Shirjaev A. N., Reduced Form of Non-linear Filtering Equations, Supplementary to Preprints of IFAC Symp. on Stochastic Control, Budapest, 1974, 59–61

[14] Gikhman I. I., Skorokhod A. V., Stokhasticheskie differentsialnye uravneniya, Naukova dumka, Kiev, 1968 | Zbl

[15] Gikhman I. I., Skorokhod A. V., Teoriya sluchainykh protsessov, t. I, Nauka, M., 1971 | MR

[16] Meyer P.-A., Martingales and Stochastic Integrals. I, Lecture Notes in Math., 284, 1972 | Zbl

[17] Pardoux E., These, L'Universite Paris Sud, d'Orsay, 1975

[18] Danford N., Shvarts Dzh., Lineinye operatory. Obschaya teoriya, IL, M., 1962