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

Gamma-type operators and the Black–Scholes semigroup

“…Combining formulas (2.3), (2.25) and (2.27), we obtain the integral representation (2.3) of the semigroup (S m (t)) t≥0 appearing in [7].…”

“…(k(n)) n≥1 being an arbitrary sequence of positive integers such that lim n→+∞ k(n) n = σ 2 t. A closer insight into the foregoing operators and semigroup may be found in [7]. In order to continue and deepen such investigation, we first note that, in a slowly different fashion, (2.2) may be rewritten as…”

“…Subsequently, other results concerning both the operators Q n and the limiting semigroup (S m (t)) t≥0 , with particular emphasis on shape-preserving properties and the asymptotic behaviour, have been obtained in [7].…”

“…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.…”

“…Finally, the last Section 4 deals with the asymptotic behaviour of Q n f (x) and, more generally, of Q k n f (x) (k ≥ 2) as x → +∞: the main result ensures the existence of an asymptote at +∞ and is strongly connected to [7,Theorem 3.1]. …”