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

*(English)*Zbl 1174.47037Summary: 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.

##### MSC:

47D06 | One-parameter semigroups and linear evolution equations |

47E05 | General theory of ordinary differential operators |

41A35 | Approximation by operators (in particular, by integral operators) |

41A36 | Approximation by positive operators |

47N10 | Applications of operator theory in optimization, convex analysis, mathematical programming, economics |

91B28 | Finance etc. (MSC2000) |

##### Keywords:

strongly continuous semigroups; differential operators; positive linear operators; Black-Scholes operator
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\textit{A. Attalienti} and \textit{I. Rasa}, Czech. Math. J. 58, No. 2, 457--467 (2008; Zbl 1174.47037)

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##### References:

[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. · Zbl 1065.41035 |

[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; Ibid. 27 (2006), 497–498. · Zbl 1065.41035 |

[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. · Zbl 1059.47045 |

[4] | F. Altomare, I. Rasa: On a class of exponential-type operators and their limit semi-groups. J. Approximation Theory 135 (2005), 258–275. · Zbl 1076.41010 |

[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. · Zbl 1174.41018 |

[6] | I. Carbone: Shape preserving properties of some positive linear operators on unbounded intervals. J. Approximation Theory 93 (1998), 140–156. · Zbl 0921.47035 |

[7] | N. El Karoui, M. Jeanblanc-Picqué, and S. E. Shreve: Robustness of the Black and Scholes formula. Math. Finance 8 (1998), 93–126. · Zbl 0910.90008 |

[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. · Zbl 0821.65065 |

[9] | R. Korn, E. Korn: Option Pricing and Portfolio Optimization. Modern Methods of Financial Mathematics. Amer. Math. Soc., Providence, 2001. · Zbl 0965.91020 |

[10] | M. Kovács: On positivity, shape, and norm-bound preservation of time-stepping methods for semigroups. J. Math. Anal. Appl. 304 (2005), 115–136. · Zbl 1068.47050 |

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

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