Francesco Altomare

Degenerate Evolution Equations in Weighted Continuous Function Spaces, Markov Processes and the Black-Scholes Equation-Part II

In this second part of the paper, through applying semigroup theory procedures, we study initial boundary problems associated with degenerate second-order differential operators of the form Lu(x) ≔ α(x) u″(x)+β(x)u′(x)+γ(x) u(x) in the framework of weighted continuous function spaces on an arbitrary real interval, when particular boundary conditions are imposed. By using the general results stated in the first part, we show that such operators, frequently occurring in Mathematical Finance, generate positive strongly continuous semigroups, which are, in turn, the transition semigroups associated with suitable Markov processes. Finally, an application to the Black-Scholes equation is discussed, as well.

On the universal convergence sets

SummaryIn this paper we study various universal convergence sets, in connection with some problem arising from the Korovkin approximation theory. This analysis is made in the context of topological vector lattices, of commutative Banach algebras, of Banach algebras with an involution and of C*-algebras. Some applications and examples are given in specific function spaces and function algebras.

On Bernstein–Schnabl Operators on the Unit Interval

In this paper we study the Bernstein-Schnabl operators associated with a continuous selection of Borel measures on the unit interval. We investigate their approximation properties by presenting several estimates of the rate of convergence in terms of suitable moduli of smoothness. We also study some shape preserving properties as well as the preservation of the convexity. Moreover we show that their iterates converge to a Markov semigroup whose generator is a degenerate second order elliptic differential operator on the unit interval. Qualitative properties of this semigroup are also investigated together with its asymptotic behaviour.

On a Class of Elliptic-Parabolic Equations on Unbounded Intervals

We study a class of degenerate elliptic second order differential operators acting on some polynomial weighted function spaces on [0,+∞[. We show that these operators are the generators of C0-semigroups of positive operators which, in turn, are the transition semigroups associated with right-continuous normal Markov processes with state space [0,+∞]. Approximation and qualitative properties of both the semigroups and the Markov processes are investigated as well. Most of the results of the paper depend on a representation of the semigroups we give in terms of powers of particular positive operators of discrete type we introduced and studied in a previous paper.

On Some Classes of Diffusion Equations and Related Approximation Problems

Of concern is a class of second-order differential operators on the unit interval. The C0-semigroup generated by them is approximated by iterates of positive linear operators that are introduced here as a modification of Bernstein operators. Finally, the corresponding stochastic differential equations are also investigated, leading, in particular to the evaluation of the asymptotic behaviour of the semigroup.

Lototsky-Schnabl operators on the unit interval and degenerate diffusion equations

Abstract The article concerns a sequence of positive linear operators on [0, 1]), called Lototsky-Schnabl operators, which are associated with a continuous function λ:[0,1]→[0,1]. We show that, if λ is a strictly positive polynomial, then there exists a positive contraction semigroup that can be expressed in terms of iterates of these operators. We give some applications of this result dealing with initial value problems associated with particular degenerate diffusion equations.

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

ABSTRACT In this paper, we present a general method to approximate the C 0–semigroups generated by special classes of one-dimensional second-order differential operators acting on weighted spaces of continuous functions by means of iterates of constructively defined positive operators. Some applications are given both for compact intervals and for unbounded intervals. In particular, an application to the Black–Scholes equation is discussed as well. Due to its length, the paper is split into two parts; the second part is published in this journal as well.

Feller Semigroups, Bernstein type Operators and Generalized Convexity Associated with Positive Projections

We study the majorizing approximation properties of both Bernstein type operators and the corresponding Feller semigroups associated with a positive projection acting on the space of all continuous functions defined on a convex compact set.

Korovkin Closures in Banach Algebras

Then the problem is, given a subset S of A and a class L of continuous linear operators on A, to determine the L Korovkin closure KorL(S) and to characterize those sets of A which have the property that KorL(S) = A. In this paper we give a short survey and some new results about this problem, and some others related to it, in the context of particular Banach algebras A and for special classes L of operators on A. More precisely, we consider function algebras on compact Hausdorff spaces and, in this case, L is the class L 1 of all linear contractions on A; moreover we study the case of C* -algebras and of the classes L + and L 1,+ of all continuous positive operators and of all positive linear contractions on A. Finally some open questions are discussed.

A generalization of Kantorovich operators for convex compact subsets

In this paper we introduce and study a new sequence of positive linear operators acting on function spaces defined on a convex compact subset. Their construction depends on a given Markov operator, a positive real number and a sequence of probability Borel measures. By considering special cases of these parameters for particular convex compact subsets we obtain the classical Kantorovich operators defined in the one-dimensional and multidimensional setting together with several of their wide-ranging generalizations scattered in the literature. We investigate the approximation properties of these operators by also providing several estimates of the rate of convergence. Finally, the preservation of Lipschitz-continuity as well as of convexity are discussed