On a class of exponential-type operators and their limit semigroups

Abstract

The paper is mainly focused upon the study of a class of second order degenerate elliptic operators on unbounded intervals.We show that these operators generate strongly continuous semigroups in suitable weighted spaces of continuous functions.Furthermore, we represent the semigroups as limits of iterates of the so-called exponential-type operators.In a particular case, starting from the stochastic differential equations associated with these operators, we also find an integral representation of the semigroup and determine its asymptotic behaviour.

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

“…P r o o f. For r = 0 and σ = 1, the proof may be found in [4,Theorem 4.4]. In the general case r 0, σ > 0, it runs essentially in the same way except for some slight changes somewhere and is therefore omitted for the sake of brevity; note that for each t 0 (compare with [4, Formula (5), p. 270])…”

“…We first provide an integral representation of (S m (t)) t 0 by using some methods developed, in a less general case, in [4] and based essentially upon the study of the stochastic differential equation associated with L. In this way we are able to determine the existence of an (oblique) asymptote of S m (t)f at +∞.…”

Convergence of iterates of genuine and ultraspherical Durrmeyer operators to the limiting semigroup: C2-estimates

“…The approximation of semigroups, as in (iii), by iterates of positive (not necessarily commuting) linear operators was intensively investigated by Francesco Altomare and his school; see [2][3][4] and the references therein. The significance of (i)-(iii) (in particular the role of the closability) is illustrated by Theorems 6.2.6 and 6.3.5 in [2].…”

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

“…For other classes of differential operators studied in the same spirit of this paper, in the framework of weighted spaces of continuous functions on [0, +∞[, we refer to [6][7][8]14] and [15]. For other classes of differential operators studied in the same spirit of this paper, in the framework of weighted spaces of continuous functions on [0, +∞[, we refer to [6][7][8]14] and [15].…”

“…In the setting of unbounded intervals, we also refer to [6][7][8], and [14] for additional results about the representation of semigroups by positive linear operators.…”