Hölder regularity for a Kolmogorov equation
Author:
Andrea Pascucci
Journal:
Trans. Amer. Math. Soc. 355 (2003), 901924
MSC (2000):
Primary 35K57, 35K65, 35K70
Published electronically:
October 1, 2002
MathSciNet review:
1938738
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: We study the interior regularity properties of the solutions to the degenerate parabolic equation,
which arises in mathematical finance and in the theory of diffusion processes.
 1.
Antonelli, F., Barucci, E., and Mancino, M. E. A Comparison result for FBSDE with Applications to Decisions Theory. Math. Methods Oper. Res. 2001, 54 (3), 407423.
 2.
Antonelli, F. and Pascucci, A. On the viscosity solutions of a stochastic differential utility problem. To appear in J. Differential Equations.
 3.
Beals, R. and Hölder estimates for pseudodifferential operators: sufficient conditions. Ann. Inst. Fourier 1979, 29 (3), 239260. MR 81c:47049
 4.
Bramanti, M. and Brandolini, L. estimates for uniformly hypoelliptic operators with discontinuous coefficients on homogeneous groups. Trans. Amer. Math. Soc. 2000, 352 (2), 781822.
 5.
Citti, G. regularity of solutions of a quasilinear equation related to the Levi operator. Ann. Scuola Norm. Sup. Pisa Cl. Sci., Serie IV 1996, 23, 483529.MR 98b:35072
 6.
Citti, G. and Montanari, A. regularity of solutions of an equation of Levi's type in . Ann. Mat. Pura Appl.(4) 2001, 180 (1), 2758. MR 2002f:35049
 7.
Citti, G., Pascucci, A., and Polidoro, S. On the regularity of solutions to a nonlinear ultraparabolic equation arising in mathematical finance. Diff. Integral Equations 2001, 14 (6), 701738. MR 2002f:35118
 8.
Citti, G., Pascucci, A., and Polidoro, S. Regularity properties of viscosity solutions of a nonHörmander degenerate equation. J. Math. Pures Appl. 2001, 80 (9), 901918.
 9.
Escobedo, M., Vazquez, J. L., and Zuazua, E. Entropy solutions for diffusionconvection equations with partial diffusivity. Trans. Amer. Math. Soc. 1994, 343 (2), 829842. MR 94h:35131
 10.
Folland, G. B. Subelliptic estimates and function spaces on nilpotent Lie groups. Ark. Mat. 1975, 13, 161207. MR 58:13215
 11.
Hörmander, L. Hypoelliptic second order differential equations. Acta Math. 1967, 119, 147171. MR 36:5526
 12.
Krylov, N. V. Hölder continuity and estimates for elliptic equations under general Hörmander's condition. Topological Methods Nonlinear Anal. 1997, 9 (2), 249258. MR 99b:35077
 13.
Lanconelli, E., Pascucci, A, and Polidoro, S. Linear and nonlinear ultraparabolic equations of Kolmogorov type arising in diffusion theory and in finance. To appear on ``Nonlinear Problems in Mathematical Physics and Related Topics Vol. II In Honor of Professor O.A. Ladyzhenskaya". International Mathematical Series, Kluwer Ed.
 14.
Lanconelli, E. and Polidoro, S. On a class of hypoelliptic evolution operators. Rend. Semin. Mat. Torino 1994, 52 (1), 2963. MR 95h:35044
 15.
Nagel A. and Stein, E. M. A new class of pseudodifferential operators. Proc. Nat. Acad. Sci. U.S.A. 1978, 75 (2), 582585. MR 58:7222
 16.
Nagel, A., Stein, E. M., and Wainger, S. Balls and metrics defined by vector fields I: Basic properties. Acta Math. 1985, 155, 103147. MR 86k:46049
 17.
Rothschild, L. P. and Stein, E. M. Hypoelliptic differential operators on nilpotent groups. Acta Math. 1977, 137, 247320. MR 55:9171
 18.
Shiryayev, A. N. (Ed.) Selected works of A. N. Kolmogorov. Vol. II. Probability theory and mathematical statistics. Kluwer Academic Publishers Group, Dordrecht, 1992, 597 pp. MR 92j:01071
 19.
Xu, C. Regularity for quasilinear secondorder subelliptic equations. Comm. Pure Appl. Math. 1992, 45, 7796. MR 93b:35042
 1.
 Antonelli, F., Barucci, E., and Mancino, M. E. A Comparison result for FBSDE with Applications to Decisions Theory. Math. Methods Oper. Res. 2001, 54 (3), 407423.
 2.
 Antonelli, F. and Pascucci, A. On the viscosity solutions of a stochastic differential utility problem. To appear in J. Differential Equations.
 3.
 Beals, R. and Hölder estimates for pseudodifferential operators: sufficient conditions. Ann. Inst. Fourier 1979, 29 (3), 239260. MR 81c:47049
 4.
 Bramanti, M. and Brandolini, L. estimates for uniformly hypoelliptic operators with discontinuous coefficients on homogeneous groups. Trans. Amer. Math. Soc. 2000, 352 (2), 781822.
 5.
 Citti, G. regularity of solutions of a quasilinear equation related to the Levi operator. Ann. Scuola Norm. Sup. Pisa Cl. Sci., Serie IV 1996, 23, 483529.MR 98b:35072
 6.
 Citti, G. and Montanari, A. regularity of solutions of an equation of Levi's type in . Ann. Mat. Pura Appl.(4) 2001, 180 (1), 2758. MR 2002f:35049
 7.
 Citti, G., Pascucci, A., and Polidoro, S. On the regularity of solutions to a nonlinear ultraparabolic equation arising in mathematical finance. Diff. Integral Equations 2001, 14 (6), 701738. MR 2002f:35118
 8.
 Citti, G., Pascucci, A., and Polidoro, S. Regularity properties of viscosity solutions of a nonHörmander degenerate equation. J. Math. Pures Appl. 2001, 80 (9), 901918.
 9.
 Escobedo, M., Vazquez, J. L., and Zuazua, E. Entropy solutions for diffusionconvection equations with partial diffusivity. Trans. Amer. Math. Soc. 1994, 343 (2), 829842. MR 94h:35131
 10.
 Folland, G. B. Subelliptic estimates and function spaces on nilpotent Lie groups. Ark. Mat. 1975, 13, 161207. MR 58:13215
 11.
 Hörmander, L. Hypoelliptic second order differential equations. Acta Math. 1967, 119, 147171. MR 36:5526
 12.
 Krylov, N. V. Hölder continuity and estimates for elliptic equations under general Hörmander's condition. Topological Methods Nonlinear Anal. 1997, 9 (2), 249258. MR 99b:35077
 13.
 Lanconelli, E., Pascucci, A, and Polidoro, S. Linear and nonlinear ultraparabolic equations of Kolmogorov type arising in diffusion theory and in finance. To appear on ``Nonlinear Problems in Mathematical Physics and Related Topics Vol. II In Honor of Professor O.A. Ladyzhenskaya". International Mathematical Series, Kluwer Ed.
 14.
 Lanconelli, E. and Polidoro, S. On a class of hypoelliptic evolution operators. Rend. Semin. Mat. Torino 1994, 52 (1), 2963. MR 95h:35044
 15.
 Nagel A. and Stein, E. M. A new class of pseudodifferential operators. Proc. Nat. Acad. Sci. U.S.A. 1978, 75 (2), 582585. MR 58:7222
 16.
 Nagel, A., Stein, E. M., and Wainger, S. Balls and metrics defined by vector fields I: Basic properties. Acta Math. 1985, 155, 103147. MR 86k:46049
 17.
 Rothschild, L. P. and Stein, E. M. Hypoelliptic differential operators on nilpotent groups. Acta Math. 1977, 137, 247320. MR 55:9171
 18.
 Shiryayev, A. N. (Ed.) Selected works of A. N. Kolmogorov. Vol. II. Probability theory and mathematical statistics. Kluwer Academic Publishers Group, Dordrecht, 1992, 597 pp. MR 92j:01071
 19.
 Xu, C. Regularity for quasilinear secondorder subelliptic equations. Comm. Pure Appl. Math. 1992, 45, 7796. MR 93b:35042
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Additional Information
Andrea Pascucci
Affiliation:
Dipartimento di Matematica, Università di Bologna, Piazza di Porta S. Donato 5, 40126 Bologna, Italy
Email:
pascucci@dm.unibo.it
DOI:
http://dx.doi.org/10.1090/S0002994702031513
PII:
S 00029947(02)031513
Received by editor(s):
June 27, 2002
Published electronically:
October 1, 2002
Additional Notes:
Investigation supported by the University of Bologna. Funds for selected research topics
Article copyright:
© Copyright 2002
American Mathematical Society
