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Shape-preserving properties and asymptotic behaviour of the semigroup generated by the Black–Scholes operator. (English) Zbl 1174.47037
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.

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)
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References:
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[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
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