The obstacle problem for a class of hypoelliptic ultraparabolic equations

Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences

Volumn 464, Issue 2089, 2008, Pages 155-176

  Di Francesco, Marco ^a   Pascucci, Andrea ^a   Polidoro, Sergio ^a  

^a UNIVERSITY OF BOLOGNA   (Italy)

Author keywords

Kolmogorov equations; Obstacle problem; Partial differential equations

Indexed keywords

AMERICAN OPTIONS; BLACK SCHOLES; DIRICHLET PROBLEMS; OBSTACLE PROBLEMS; STRONG SOLUTIONS;

BOUNDARY VALUE PROBLEMS;

EID: 47149084068     PISSN: 13645021     EISSN: 14712946     Source Type: Journal    
DOI: 10.1098/rspa.2007.0090     Document Type: Article

References (35)

1. Barles, G. 1997 Convergence of numerical schemes for degenerate parabolic equations arising in finance theory. In Numerical methods in finance (eds L. C. G. Rogers & D. Talay), pp. 1-21, Cambridge, UK: Cambridge University Press.
2. Barraquand, J. & Pudet, T. 1996 Pricing of American path-dependent contingent claims. Math. Finance 6, 17-51. (doi:10.1111/j.1467- 9965.1996.tb00111.x)
3. Barucci, E., Polidoro, S. & Vespri, V. 2001 Some results on partial differential equations and Asian options. Math. Models Methods Appl. Sci. 11, 475-497. (doi:10.1142/S0218202501000945)
4. Bensoussan, A. 1984 On the theory of option pricing. Acta Appl. Math. 2, 139-158.
5. Bensoussan, A. & Lions, J.-L. 1982 Applications of variational inequalities in stochastic control, vol. 12. Studies in mathematics and its applications. Amsterdam: North-Holland.
6. p estimates for some ultraparabolic operators with discontinuous coefficients. J. Math. Anal Appl. 200, 332-354. (doi:10.1006/jmaa.1996.0209)
7. Crandall, M. G., Ishii, H. & Lions, P.-L. 1992 User's guide to viscosity solutions of second order partial differential equations. Bull. Am. Math. Soc. (N.S.) 27, 1-67.
8. Dai, M. & Kwok, Y. K. 2006 Characterization of optimal stopping regions of American Asian and lookback options. Math. Finance 16, 63-82. (doi:10.1111/j.1467-9965.2006.00261.x)
9. Di Francesco, M. & Pascucci, A. 2004 On the complete model with stochastic volatility by Hobson and Rogers. Proc. R. Soc. A 460, 3327-3338. (doi:10.1098/rspa.2004.1370)
10. Di Francesco, M. & Polidoro, S. 2006 Schauder estimates, Harnack inequality and Gaussian lower bound for Kolmogorov type operators in non-divergence form. Adv. Differ. Eqns 11, 1261-1320.
11. Folland, G. B. 1975 Subelliptic estimates and function spaces on nilpotent Lie groups. Ark. Mat. 13, 161-207. (doi:10.1007/ BF02386204)
12. Foschi, P. & Pascucci A. In press. Path dependent volatility. Decis. Econ. Finance.
13. Friedman, A. 1964 Partial differential equations of parabolic type. Englewood Cliffs, NJ: Prentice-Hall, Inc.
14. Friedman, A. 1975 Parabolic variational inequalities in one space dimension and smoothness of the free boundary. J. Funct. Anal. 18, 151-176. (doi: 10.1016/0022-1236(75)90022-1)
15. Friedman, A. 1982 Variational principles and free-boundary problems: Pure and applied mathematics. New York, NY: Wiley/A Wiley-Interscience Publication.
16. Friedman, A. 1988 Variational principles and free-boundary problems, 2nd edn. Malabar, FL: Robert E. Krieger Publishing Co., Inc.
17. Gilbarg, D. & Trudinger, N. S. 1977 Elliptic partial differential equations of second order, vol. 224. Grundlehren der mathematischen Wissenschaften. Berlin, Germany: Springer.
18. Hansen, A. T. & Jorgensen, P. L. 2000 Analytical valuation of American-style Asian options. Manag. Sci. 46, 1116-1136. (doi:10.1287/mnsc.46.8.1116.12027)
19. Hobson, D. G. & Rogers, L. C. G. 1998 Complete models with stochastic volatility. Math. Finance 8, 27-48. (doi:10.1111/1467-9965.00043)
20. Hörmander, L. 1967 Hypoelliptic second order differential equations. Acta Math. 119, 147-171. (doi:10.1007/BF02392081)
21. Jaillet, P., Lamberton, D. & Lapeyre, B. 1990 Variational inequalities and the pricing of American options, Acta Appl. Math. 21, 263-289. (doi:10.1007/BF00047211)
22. Jiang, L. & Dai, M. 2004 Convergence of binomial tree methods for European/American path-dependent options. SIAM J. Numer. Anal. 42, 1094-1109. (doi:10.1137/S0036142902414220)
23. Karatzas, I. 1988 On the pricing of American options. Appl. Math. Optim. 17, 37-60. (doi:10.1007/BF01448358)
24. Kinderlehrer, D. & Stampacchia, G. 2000 An introduction to variational inequalities and their applications: Classics in applied mathematics, vol. 31. Philadelphia, PA: Society for Industrial and Applied Mathematics (SIAM).
25. Lanconelli, E. & Polidoro, S. 1994 On a class of hypoelliptic evolution operators. Rend. Sem. Mat. Univ. Politec. Torino 52, 29-63.
26. Manfredini, M. 1997 The Dirichlet problem for a class of ultraparabolic equations. Adv. Differ. Eqns 2, 831-866.
27. Manfredini, M. & Polidoro, S. 1998 Interior regularity for weak solutions of ultraparabolic equations in divergence form with discontinuous coefficients. Boll. Unione Mat. Ital. Sez. B Artie. Ric. Mat. 1, 651-675.
28. Marcozzi, M. D. 2003 On the valuation of Asian options by variational methods. SIAM J. Sci. Comput. 24, 1124-1140. (doi:10.1137/ S1064827501388169)
29. Meyer, G. H. 2000 On pricing American and Asian options with PDE methods. Acta Math. Univ. Comenian. (N.S.) 70, 153-165.
30. Nagel, A., Stein, E. M. & Wainger, S. 1985 Balls and metrics defined by vector fields. I. Basic properties. Acta Math. 155, 103-147. (doi:10.1007/BF02392539)
31. Pascucci, A. In press. Free boundary and optimal stopping problems for American Asian options. Finance Stoch.
32. Peskir, G. & Shiryaev, A. 2006 Optimal stopping and free-boundary problems. Lectures in Mathematics. ETH Basel: Zürich. Birkhäuser: Germany, xxii, pp. 500.
33. Polidoro, S. &: Ragusa, M. A. 1998 Sobolev-Morrey spaces related to an ultraparabolic equation. Manuscr. Math. 96, 371-392. (doi:10.1007/s002290050072)
34. Rogers, L. & Shi, Z. 1995 The value of an Asian option. J. Appl. Probab. 32, 1077-1088. (doi:10.2307/3215221)
35. Rothschild, L. P. & Stein, E. M. 1976 Hypoelliptic differential operators and nilpotent groups. Acta Math. 137, 247-320. (doi:10.1007/BF02392419)