An analytic approach to purely nonlocal Bellman equations arising in models of stochastic control

Journal of Differential Equations

Volumn 236, Issue 1, 2007, Pages 29-56

  Abels, H ^a   Kassmann, M ^b  

^a MAX PLANCK INSTITUTE FOR MATHEMATICS IN THE SCIENCES   (Germany)

^b UNIVERSITY OF BONN   (Germany)

Author keywords

Bellman equation; Fully nonlinear equation; Integro differential operator; Markov jump process; Stochastic control

Indexed keywords

EID: 33947599952     PISSN: 00220396     EISSN: 10902732     Source Type: Journal    
DOI: 10.1016/j.jde.2006.12.013     Document Type: Article

References (25)

1. Adams R.A. Sobolev Spaces (1975), Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York
2. Alvarez O., and Tourin A. Viscosity solutions of nonlinear integro-differential equations. Ann. Inst. H. Poincaré Anal. Non Linéaire 13 3 (1996) 293-317
3. Bogdan K., Burdzy K., and Chen Z.-Q. Censored stable processes. Probab. Theory Related Fields 127 1 (2003) 89-152
4. Brézis H., and Evans L.C. A variational inequality approach to the Bellman-Dirichlet equation for two elliptic operators. Arch. Ration. Mech. Anal. 71 1 (1979) 1-13
5. Bass R., and Kassmann M. Harnack inequalities for non-local operators of variable order. Trans. Amer. Math. Soc. 357 2 (2005) 837-850
6. Bass R., and Kassmann M. Hölder continuity of harmonic functions with respect to operators of variable order. Comm. Partial Differential Equations 30 (2005) 1249-1259
7. Bergh J., and Löfström J. Interpolation Spaces (1976), Springer, Berlin
8. Caffarelli L.A., and Cabré X. Fully Nonlinear Elliptic Equations. American Mathematical Society Colloquium Publications vol. 43 (1995), American Mathematical Society, Providence, RI
9. Crandall M.G., Ishii H., and Lions P.-L. User's guide to viscosity solutions of second order partial differential equations. Bull. Amer. Math. Soc. (N.S.) 27 1 (1992) 1-67
10. Evans L.C., and Friedman A. Optimal stochastic switching and the Dirichlet problem for the Bellman equation. Trans. Amer. Math. Soc. 253 (1979) 365-389
11. Evans L.C. Classical solutions of the Hamilton-Jacobi-Bellman equation for uniformly elliptic operators. Trans. Amer. Math. Soc. 275 1 (1983) 245-255
12. Fukushima M., Ōshima Y., and Takeda M. Dirichlet Forms and Symmetric Markov Processes. de Gruyter Studies in Mathematics vol. 19 (1994), de Gruyter, Berlin
13. Gimbert F., and Lions P.-L. Existence and regularity results for solutions of second-order, elliptic integro-differential operators. Ricerche Mat. 33 2 (1984) 315-358
14. Hellmann M. Über die voll nichtlineare Bellman-Gleichung mit subquadratischem Kostenfunktional (2001), Universität Bonn, Diplomarbeit
15. Jakobsen E.R., and Karlsen K.H. Continuous dependence estimates for viscosity solutions of integro-PDEs. J. Differential Equations 212 2 (2005) 278-318
16. Krylov N.V. Controlled Diffusion Processes. Applications of Mathematics vol. 14 (1980), Springer, New York
17. Krylov N.V. Fully nonlinear second order elliptic equations: Recent development. Ann. Sc. Norm. Super. Pisa Cl. Sci. (4) 25 3-4 (1998) 569-595 1997
18. Krylov N.V., and Safonov M.V. A property of the solutions of parabolic equations with measurable coefficients. Izv. Akad. Nauk SSSR Ser. Mat. 44 1 (1980) 161-175 239
19. Morrey Jr. C.B. Multiple Integrals in the Calculus of Variations. Die Grundlehren der mathematischen Wissenschaften, Band 130 (1966), Springer, New York
20. Mikulevičius R., and Pragarauskas H. On the uniqueness of a solution to the Bellman equation in Sobolev's classes. Litovsk. Mat. Sb. 31 4 (1991) 646-664
21. Mikulevičius R., and Pragarauskas G. Classical solutions of boundary value problems for some nonlinear integro-differential equations. Liet. Mat. Rink. 34 3 (1994) 347-361
22. Mikulevičius R., and Pragarauskas G. On the existence of viscosity solutions to boundary value problems for integrodifferential Bellman equation. Probability Theory and Mathematical Statistics. Tokyo, 1995 (1996), World Sci. Publishing 327-342
23. Menaldi J.L., and Robin M. Ergodic control of reflected diffusions with jumps. Appl. Math. Optim. 35 2 (1997) 117-137
24. Silvestre L. Hölder estimates for solutions of integro differential equations like the fractional Laplace. Indiana Univ. Math. J. 55 3 (2006) 1155-1174
25. Simon J. Sobolev, Besov and Nikolskii fractional spaces: Imbeddings and comparisons for vector valued spaces on an interval. Ann. Mat. Pura Appl. (4) 157 (1990) 117-148