Geodesic
Shape-preserving properties and asymptotic behaviour of the semigroup generated by the Black-Scholes operator
Czechoslovak Mathematical Journal, Tome 58 (2008) no. 2, pp. 457-467 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

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.
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.
Classification : 41A35, 41A36, 47D06, 47E05, 47N10, 91B28, 91Gxx
Keywords: strongly continuous semigroups; differential operators; positive linear operators; Black-Scholes operator 
@article{CMJ_2008_58_2_a10,
     author = {Attalienti, Antonio and Rasa, Ioan},
     title = {Shape-preserving properties and asymptotic behaviour of the semigroup generated by the {Black-Scholes} operator},
     journal = {Czechoslovak Mathematical Journal},
     pages = {457--467},
     year = {2008},
     volume = {58},
     number = {2},
     mrnumber = {2411101},
     zbl = {1174.47037},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/CMJ_2008_58_2_a10/}
}
TY  - JOUR
AU  - Attalienti, Antonio
AU  - Rasa, Ioan
TI  - Shape-preserving properties and asymptotic behaviour of the semigroup generated by the Black-Scholes operator
JO  - Czechoslovak Mathematical Journal
PY  - 2008
SP  - 457
EP  - 467
VL  - 58
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/CMJ_2008_58_2_a10/
LA  - en
ID  - CMJ_2008_58_2_a10
ER  - 
%0 Journal Article
%A Attalienti, Antonio
%A Rasa, Ioan
%T Shape-preserving properties and asymptotic behaviour of the semigroup generated by the Black-Scholes operator
%J Czechoslovak Mathematical Journal
%D 2008
%P 457-467
%V 58
%N 2
%U http://geodesic.mathdoc.fr/item/CMJ_2008_58_2_a10/
%G en
%F CMJ_2008_58_2_a10
Attalienti, Antonio; Rasa, Ioan. Shape-preserving properties and asymptotic behaviour of the semigroup generated by the Black-Scholes operator. Czechoslovak Mathematical Journal, Tome 58 (2008) no. 2, pp. 457-467. http://geodesic.mathdoc.fr/item/CMJ_2008_58_2_a10/

[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. | DOI | MR

[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. | DOI | MR

[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. | DOI | MR

[4] F.  Altomare, I.  Rasa: On a class of exponential-type operators and their limit semigroups. J.  Approximation Theory 135 (2005), 258–275. | DOI | MR

[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

[6] I.  Carbone: Shape preserving properties of some positive linear operators on unbounded intervals. J.  Approximation Theory 93 (1998), 140–156. | DOI | MR | Zbl

[7] N.  El  Karoui, M.  Jeanblanc-Picqué, and S. E.  Shreve: Robustness of the Black and Scholes formula. Math. Finance 8 (1998), 93–126. | DOI | MR

[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. | DOI | MR

[9] R.  Korn, E.  Korn: Option Pricing and Portfolio Optimization. Modern Methods of Financial Mathematics. Amer. Math. Soc., Providence, 2001. | MR

[10] M.  Kovács: On positivity, shape, and norm-bound preservation of time-stepping methods for semigroups. J.  Math. Anal. Appl. 304 (2005), 115–136. | DOI | MR

[11] B.Øksendal: Stochastic Differential Equations. Fourth Edition. Springer-Verlag, New York, 1995.