Hölder regularity for a Kolmogorov equation

Author:
Andrea Pascucci

Journal:
Trans. Amer. Math. Soc. **355** (2003), 901-924

MSC (2000):
Primary 35K57, 35K65, 35K70

Published electronically:
October 1, 2002

MathSciNet review:
1938738

Full-text 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), 407-423.**2.**Antonelli, F. and Pascucci, A. On the viscosity solutions of a stochastic differential utility problem. To appear in J. Differential Equations.**3.**Richard Beals,*𝐿^{𝑝} and Hölder estimates for pseudodifferential operators: sufficient conditions*, Ann. Inst. Fourier (Grenoble)**29**(1979), no. 3, vii, 239–260 (English, with French summary). MR**552967****4.**Bramanti, M. and Brandolini, L. estimates for uniformly hypoelliptic operators with discontinuous coefficients on homogeneous groups. Trans. Amer. Math. Soc.**2000**,*352*(2), 781-822.**5.**G. Citti,*𝐶^{∞} regularity of solutions of a quasilinear equation related to the Levi operator*, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4)**23**(1996), no. 3, 483–529. MR**1440031****6.**Giovanna Citti and Annamaria Montanari,*𝐶^{∞} regularity of solutions of an equation of Levi’s type in 𝑅²ⁿ⁺¹*, Ann. Mat. Pura Appl. (4)**180**(2001), no. 1, 27–58. MR**1848050**, 10.1007/s10231-001-8196-z**7.**Giovanna Citti, Andrea Pascucci, and Sergio Polidoro,*On the regularity of solutions to a nonlinear ultraparabolic equation arising in mathematical finance*, Differential Integral Equations**14**(2001), no. 6, 701–738. MR**1826957****8.**Citti, G., Pascucci, A., and Polidoro, S. Regularity properties of viscosity solutions of a non-Hörmander degenerate equation. J. Math. Pures Appl.**2001**,*80*(9), 901-918.**9.**M. Escobedo, J. L. Vázquez, and Enrike Zuazua,*Entropy solutions for diffusion-convection equations with partial diffusivity*, Trans. Amer. Math. Soc.**343**(1994), no. 2, 829–842. MR**1225573**, 10.1090/S0002-9947-1994-1225573-2**10.**G. B. Folland,*Subelliptic estimates and function spaces on nilpotent Lie groups*, Ark. Mat.**13**(1975), no. 2, 161–207. MR**0494315****11.**Lars Hörmander,*Hypoelliptic second order differential equations*, Acta Math.**119**(1967), 147–171. MR**0222474****12.**N. V. Krylov,*Hölder continuity and 𝐿_{𝑝} estimates for elliptic equations under general Hörmander’s condition*, Topol. Methods Nonlinear Anal.**9**(1997), no. 2, 249–258. MR**1491846****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.**E. Lanconelli and S. Polidoro,*On a class of hypoelliptic evolution operators*, Rend. Sem. Mat. Univ. Politec. Torino**52**(1994), no. 1, 29–63. Partial differential equations, II (Turin, 1993). MR**1289901****15.**Alexander Nagel and E. M. Stein,*A new class of pseudo-differential operators*, Proc. Nat. Acad. Sci. U.S.A.**75**(1978), no. 2, 582–585. MR**0487603****16.**Alexander Nagel, Elias M. Stein, and Stephen Wainger,*Balls and metrics defined by vector fields. I. Basic properties*, Acta Math.**155**(1985), no. 1-2, 103–147. MR**793239**, 10.1007/BF02392539**17.**Linda Preiss Rothschild and E. M. Stein,*Hypoelliptic differential operators and nilpotent groups*, Acta Math.**137**(1976), no. 3-4, 247–320. MR**0436223****18.**A. N. Kolmogorov,*Selected works. Vol. II*, Mathematics and its Applications (Soviet Series), vol. 26, Kluwer Academic Publishers Group, Dordrecht, 1992. Probability theory and mathematical statistics; With a preface by P. S. Aleksandrov; Translated from the Russian by G. Lindquist; Translation edited by A. N. Shiryayev [A. N. Shiryaev]. MR**1153022****19.**Chao Jiang Xu,*Regularity for quasilinear second-order subelliptic equations*, Comm. Pure Appl. Math.**45**(1992), no. 1, 77–96. MR**1135924**, 10.1002/cpa.3160450104

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC (2000):
35K57,
35K65,
35K70

Retrieve articles in all journals with MSC (2000): 35K57, 35K65, 35K70

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:
https://doi.org/10.1090/S0002-9947-02-03151-3

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