APPROXIMATION BY POSITIVE OPERATORS OF THE C₀–SEMIGROUPS ASSOCIATED WITH ONE-DIMENSIONAL DIFFUSION EQUATIONS: PART II

Abstract

In this second part of the paper, we state some general conditions that guarantee that the C 0 -semigroups generated by special classes of one-dimensional second-order differential operators acting on weighted spaces of continuous functions on an arbitrary real interval can be represented as limits of iterates of the positive linear operators constructively defined according to the method developed in the first part. The particular case where the interval is compact is treated in full detail. Finally, an application to the Black-Scholes equation, which appears in some models arising from mathematical finance, is discussed as well.

Gamma-type operators and the Black–Scholes semigroup

“…Subsequently, in [3] (see also [4]), aiming at representing explicitly (S m (t)) t≥0 in the spirit of Altomare's theory, the authors introduced certain perturbed Post-Widder operators defined by…”

“…It goes without saying that representation formula (2.3) plays a crucial role in this kind of approach, falling within a general scheme of investigation of the use of iterates of positive linear operators in the study of evolution problems, initiated by Karlin and Ziegler in [17] and continued and developed originally and extensively, in different contexts, by Altomare and his school: without attempting to be exhaustive in this respect, we confine ourselves to citing [6,3,5,7] and the references quoted therein.…”

“…The aim of the present paper is to deepen the study of a particular sequence of positive linear operators, denoted by Q n and introduced in [3,4] as perturbed Post-Widder operators, in the setting of weighted continuous function spaces.…”

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

“…In the same framework, in [1] and [2] the authors, according with a general scheme of investigation of the interplay between constructive approximation processes and degenerate evolution problems (see, for instance, [1]- [5] and many of the references quoted therein), have shown that (S m (t)) t 0 may be written down in terms of suitable Post-Widder-type operators Q n , as described in (2.2).…”