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

Multiplicative perturbations of the Laplacian and related approximation problems

“…, N . Since pr 2 i ∈ D m (A) and since it is convex, by Proposition 5.2 we obtain that The second inequality of (5.9) is a consequence of Theorem 4.1, part (3). In order to prove the last claim, we preliminary observe that, given n ≥ 1 and p ≥ 1, since the functional for every f ∈ E 0 m .…”

“…is the pre-generator (i.e., it is closable and its closure is the generator) of a positive C 0 -semigroup on C w 0 (X ) that leaves invariant the space C 0 (X ) and whose relevant restriction to C 0 (X ) is a Feller semigroup (i.e., a positive and contractive C 0 -semigroup) on (C w 0 (X ), · ∞ ) (for more details on the theory of C 0 -semigroups we refer, e.g., to [18,16]; as regards C 0 -transition functions and Markov processes we refer to [3,Section 1], [16,20]). …”

“…The proof essentially rests on Proposition 2.3 and Theorem 2.10 of [3]. For the convenience of the reader, we sketch below more details.…”

“…Since T (t) = T (t) on C 0 (X ) (t ≥ 0), we then infer that T (t) = T (t) on C w 0 (X ). Moreover, the corresponding transition functions and Markov processes are the same (see also [3,Remark 2.11]). …”

Steklov operators and semigroups in weighted spaces of continuous real functions

“…There is a wide literature on this subject; we only refer to the pioneer paper [1] and to [3,Ch. 6] for a general treatment; in the setting of weighted spaces of continuous functions some recent results have been obtained in [10,5,6,2,4]. Steklov operators are particularly suitable in this context since they can be used both in the setting of unbounded than bounded real intervals and further their iterates can be easily evaluated.…”