Shape-preserving properties and asymptotic behaviour of the semigroup generated by the Black-Scholes operator

Antonio Attalienti; Ioan Rasa

Czechoslovak Mathematical Journal (2008)

  • Volume: 58, Issue: 2, page 457-467
  • ISSN: 0011-4642

Abstract

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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.

How to cite

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Attalienti, Antonio, and Rasa, Ioan. "Shape-preserving properties and asymptotic behaviour of the semigroup generated by the Black-Scholes operator." Czechoslovak Mathematical Journal 58.2 (2008): 457-467. <http://eudml.org/doc/31221>.

@article{Attalienti2008,
abstract = {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.},
author = {Attalienti, Antonio, Rasa, Ioan},
journal = {Czechoslovak Mathematical Journal},
keywords = {strongly continuous semigroups; differential operators; positive linear operators; Black-Scholes operator; strongly continuous semigroups; differential operators; positive linear operators; Black-Scholes operator},
language = {eng},
number = {2},
pages = {457-467},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Shape-preserving properties and asymptotic behaviour of the semigroup generated by the Black-Scholes operator},
url = {http://eudml.org/doc/31221},
volume = {58},
year = {2008},
}

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
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 58
IS - 2
SP - 457
EP - 467
AB - 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.
LA - eng
KW - strongly continuous semigroups; differential operators; positive linear operators; Black-Scholes operator; strongly continuous semigroups; differential operators; positive linear operators; Black-Scholes operator
UR - http://eudml.org/doc/31221
ER -

References

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  8. 10.1007/BF01876627, Period. Math. Hung. 30 (1995), 135–139. (1995) MR1326774DOI10.1007/BF01876627
  9. Option Pricing and Portfolio Optimization. Modern Methods of Financial Mathematics, Amer. Math. Soc., Providence, 2001. (2001) MR1802499
  10. 10.1016/j.jmaa.2004.09.069, J.  Math. Anal. Appl. 304 (2005), 115–136. (2005) MR2124652DOI10.1016/j.jmaa.2004.09.069
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