| Title:
|
Shape-preserving properties and asymptotic behaviour of the semigroup generated by the Black-Scholes operator (English) |
| Author:
|
Attalienti, Antonio |
| Author:
|
Rasa, Ioan |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
58 |
| Issue:
|
2 |
| Year:
|
2008 |
| Pages:
|
457-467 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
The paper is devoted to a careful analysis of the shape-preserving properties of the strongly continuous semigroup generated by a particular second-order differential operator, with particular emphasis on the preservation of higher order convexity and Lipschitz classes. In addition, the asymptotic behaviour of the semigroup is investigated as well. The operator considered is of interest, since it is a unidimensional Black-Scholes operator so that our results provide qualitative information on the solutions of classical problems in option pricing theory in Mathematical Finance. (English) |
| Keyword:
|
strongly continuous semigroups |
| Keyword:
|
differential operators |
| Keyword:
|
positive linear operators |
| Keyword:
|
Black-Scholes operator |
| MSC:
|
41A35 |
| MSC:
|
41A36 |
| MSC:
|
47D06 |
| MSC:
|
47E05 |
| MSC:
|
47N10 |
| MSC:
|
91B28 |
| MSC:
|
91Gxx |
| idZBL:
|
Zbl 1174.47037 |
| idMR:
|
MR2411101 |
| . |
| Date available:
|
2009-09-24T11:56:00Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/128269 |
| . |
| Reference:
|
[1] F. Altomare, R. Amiar: Approximation by positive operators of the $C_0$-semigroups associated with one-dimensional diffusion equations. Part II.Numer. Funct. Anal. Optimiz. 26 (2005), 17–33. MR 2128742, 10.1081/NFA-200051623 |
| Reference:
|
[2] F. Altomare, R. Amiar: Corrigendum to: Approximation by positive operators of the $C_0$-semigroups associated with one-dimensional diffusion equations, Part II.Numer. Funct. Anal. Optimiz. 26 (2005), 17–33. MR 2128742, 10.1081/NFA-200051623 |
| Reference:
|
[3] F. Altomare, A. Attalienti: Degenerate evolution equations in weighted continuous function spaces, Markov processes and the Black-Scholes equation. Part II.Result. Math. 42 (2002), 212–228. MR 1946741, 10.1007/BF03322851 |
| Reference:
|
[4] F. Altomare, I. Rasa: On a class of exponential-type operators and their limit semigroups.J. Approximation Theory 135 (2005), 258–275. MR 2158534, 10.1016/j.jat.2005.05.006 |
| Reference:
|
[5] A. Attalienti, I. Rasa: Total positivity: An application to positive linear operators and to their limiting semigroups.Rev. Anal. Numer. Theor. Approx. 36 (2007), 51–66. MR 2499632 |
| Reference:
|
[6] I. Carbone: Shape preserving properties of some positive linear operators on unbounded intervals.J. Approximation Theory 93 (1998), 140–156. Zbl 0921.47035, MR 1612802, 10.1006/jath.1997.3134 |
| Reference:
|
[7] N. El Karoui, M. Jeanblanc-Picqué, and S. E. Shreve: Robustness of the Black and Scholes formula.Math. Finance 8 (1998), 93–126. MR 1609962, 10.1111/1467-9965.00047 |
| Reference:
|
[8] I. Faragó, T. Pfeil: Preserving concavity in initial-boundary value problems of parabolic type and in its numerical solution.Period. Math. Hung. 30 (1995), 135–139. MR 1326774, 10.1007/BF01876627 |
| Reference:
|
[9] R. Korn, E. Korn: Option Pricing and Portfolio Optimization. Modern Methods of Financial Mathematics.Amer. Math. Soc., Providence, 2001. MR 1802499 |
| Reference:
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[10] M. Kovács: On positivity, shape, and norm-bound preservation of time-stepping methods for semigroups.J. Math. Anal. Appl. 304 (2005), 115–136. MR 2124652, 10.1016/j.jmaa.2004.09.069 |
| Reference:
|
[11] B.Øksendal: Stochastic Differential Equations. Fourth Edition.Springer-Verlag, New York, 1995. |
| . |
