Model problem for integro-differential zakai equation with discontinuous observation processes

Applied Mathematics and Optimization

Volumn 64, Issue 1, 2011, Pages 37-69

  Mikulevicius, R ^a   Pragarauskas, H ^b  

^a UNIVERSITY OF SOUTHERN CALIFORNIA   (United States)

^b VILNIUS UNIVERSITY   (Lithuania)

Author keywords

Nonlinear filtering of jump processes; Stochastic H lder spaces; Stochastic parabolic integro differential equations

Indexed keywords

CAUCHY PROBLEMS; EXISTENCE AND UNIQUENESS; FILTERING PROBLEMS; JUMP PROCESS; MODEL PROBLEMS; SIGNAL PROCESS; ZAKAI EQUATION;

DIFFERENTIAL EQUATIONS; HOLMIUM; INTEGRODIFFERENTIAL EQUATIONS; MARKOV PROCESSES; SIGNAL FILTERING AND PREDICTION; STOCHASTIC SYSTEMS;

NONLINEAR EQUATIONS;

EID: 80052490732     PISSN: 00954616     EISSN: 14320606     Source Type: Journal    
DOI: 10.1007/s00245-010-9131-8     Document Type: Article

References (18)

1. Applebaum, D.: Levy Processes and Stochastic Calculus. Cambridge University Press, Cambridge (2004)
2. Bergh, J., Löfstrom, J.: Interpolation Spaces. An Introduction. Springer, Berlin (1976)
3. Caffarelli, L., Silvestre, L.: An extension problem related to the fractional Laplacian. Commun. Partial Differ. Equ. 32(7-9), 1245-1260
4. Caffarelli, L., Silvestre, L.: Regularity theory for fully nonlinear integro-differential equations. Commun. Pure Appl. Math. 62, 597-638 (2009)
5. Caffarrelli, L., Vasseur, A.: Drift diffusion equations with fractional diffusion and the quasigeostrophic equation. Ann. Math. 171(3), 1903-1930 (2010)
6. Dellacherie, C., Meyer, P.: Probabilities and Potential. North-Holland Mathematics Studies, vol. 29. North-Holland, Amsterdam (1978)
7. Grigelionis, B.: Reduced stochastic equations of the nonlinear filtering of random processes. Lith. Math. J. 16, 348-358 (1976)
8. Jacod, J.: Calcul Stochastique et Problemes de Martingales. Lecture Notes in Mathematics, vol. 714. Springer, Berlin (1979)
9. Krylov, N.V., Rozovskii, B.L.: On the Cauchy problem for linear partial differential equations. Math. USSR Izv. 11, 1267-1284 (1977)
10. Krylov, N.V.: On Lp theory of stochastic partial differential equations. SIAM J. Math. Anal. 27, 313-340 (1996)
11. Mikulevicius, R., Pragarauskas, H.: On the Cauchy problem for certain integro-differential operators in Sobolev and Hölder spaces. Lith. Math. J. 32, 238-264 (1992)
12. Mikulevicius, R., Pragarauskas, H.: On Hölder solutions of the integro-differential Zakai equation. Stochastic Process. Appl. 119, 3319-3355 (2009)
13. Mikulevicius, R.: On the Cauchy problem for parabolic SPDEs in Hölder classes. Ann. Probab. 28, 74-108 (2000)
14. Mikulevicius, R.: Properties of solutions of stochastic differential equations. Lith. Math. J. 23, 367-376 (1983)
15. Pardoux, E.: Equations aux derivées partielles stochastiques non linéaires monotones. Étude de solutions de type Itô: Thèse. Université de Paris Sud, Orsay (1975)
16. Rozovskii, B.L.: On stochastic partial differential equations. Mat. Sb. 96, 314-341 (1975)
17. Rozovskii, B.L.: Stochastic Evolution Systems. Kluwer Academic, Norwell (1990)
18. Zakai, M.: On the optimal filtering of diffusion processes. Z. Wahrsch. 11, 230-243 (1969)