Gisèle Ruiz Goldstein

Chaotic solution for the Black-Scholes equation

The Black-Scholes semigroup is studied on spaces of continuous functions on (0,∞) which may grow at both 0 and at ∞, which is important since the standard initial value is an unbounded function. We prove that in the Banach spaces Y s,τ := {u ∈ C((0,∞)) : lim x→∞ u(x) 1 + xs = 0, lim x→0 u(x) 1 + x−τ = 0} with norm ‖u‖Y s,τ = sup x>0 ∣ ∣ ∣ u(x) (1+xs)(1+x−τ ) ∣ ∣ ∣ 1, τ ≥ 0 with sν > 1, where √ 2ν is the volatility. The proof relies on the Godefroy-Shapiro hypercyclicity criterion.

Degenerate Second Order Differential Operators Generating Analytic Semigroups inLp andW1,p

We deal with the problem of analyticity for the semigroup generated by the second order differential operator Au ≔ αu″ + βu′ (or by some restrictions of it) in the spaces Lp(0, 1), with or without weight, and in W1,p(0, 1), 1 0 in (0, 1), and the domain of A is determined by the generalized Neumann boundary conditions and by Wentzell boundary conditions.

Weighted Hardy's inequality and the Kolmogorov equation perturbed by an inverse-square potential

In this article, we give necessary and sufficient conditions for the existence of a weak solution of a Kolmogorov equation perturbed by an inverse-square potential. More precisely, using a weighted Hardys inequality with respect to an invariant measure μ, we show the existence of the semigroup solution of the parabolic problem corresponding to a generalized Ornstein–Uhlenbeck operator perturbed by an inverse-square potential in L 2(ℝ N , μ). In the case of the classical Ornstein–Uhlenbeck operator we obtain nonexistence of positive exponentially bounded solutions of the parabolic problem if the coefficient of the inverse-square function is too large.

Generalized Wentzell Boundary Conditions and Analytic Semigroups in C [0, 1]

In [4], we introduced for the first time the so-called generalized Wentzell boundary conditions for some classes of linear, or nonlinear, second order differential operator with domain in the space C[0, 1] of all real-valued continuous functions on [0, 1]. There we proved generation results which extended substantially those referred to Dirichlet, Neumann, Robin and Wentzell boundary conditions.

Nonlinear parabolic equations with singular coefficient and critical exponent

We are concerned with the absence of positive solutions of the following nonlinear problems, Here Ω is a bounded domain in with smooth boundary, 0

The Laplacian with Generalized Wentzell Boundary Conditions

The regularity of the solutions of the heat equation

The Favard Class for a Parabolic Problem with Wentzell Boundary Conditions

\frac{{\partial u}}{{\partial t}} = \Delta u

Overdamping and energy decay for abstract wave equations with strong damping

with suitable boundary conditions in different types of function spaces is an impor-tant issue in many applications to problems coming from Physics and Engineering.