It is shown, using the Borwein–Preiss variational principle that for every continuous convex function f on a weakly compactly generated space X, every x0 ∈ X and every weakly compact convex symmetric set K such that spanK = X, there is a point of Gâteaux differentiability of f in x0 +K. This extends a Klee’s result for separable spaces. The well-known Mazur’s theorem says that a continuous conv...

In this paper, we consider the ruin probabilities (caused by oscillation or by a claim) of the classical risk process perturbed by diffusion and the risk process with return on investments. We will prove their twice continuous differentiability and derive the integro-differential equations satisfied by them. We will present the explicit expressions for them when the claims are exponentially dis...

In this paper we study the behavior of the local solutions of perturbed variational inequalities, governed by perturbations to both the variational inequality function and the feasible region. Assuming appropriate second-order and regularity conditions, we show that the perturbed local solution set is nonempty, Lipschitz continuous, and directionally differentiable. Even when the directional di...

Continuing research in [13] and [14] on well-posedness of the optimal time control problem with a constant convex dynamics (in a Hilbert space), we adapt one of the regularity conditions obtained there to a slightly more general problem, where nonaffine additive term appears. We prove existence and uniqueness of a minimizer in this problem as well as continuous differentiability of the value fu...

We prove pathwise uniqueness for stochastic differential equations driven by non-degenerate symmetric α-stable Lévy processes with values in Rd having a bounded and β-Hölder continuous drift term. We assume β > 1− α/2 and α ∈ [1, 2). The proof requires analytic regularity results for the associated integro-differential operators of Kolmogorov type. We also study differentiability of solutions w...

This paper is concerned with the study of envelope theorems for finite choice sets. More specifically, we consider the problem of differentiability of the value function arising out of the maximization of a parametrized objective function, when the set of alternatives is non-empty and finite. The parameter is confined to the closed interval [0,1] and for each alternative, the objective function...

We present a short elementary proof of the Lagrange multiplier theorem for equality-constrained optimization. Most proofs in the literature rely on advanced analysis concepts such as the implicit function theorem, whereas elementary proofs tend to be long and involved. By contrast, our proof uses only basic facts from linear algebra, the definition of differentiability, the critical-point condi...

In this paper we study subdivision schemes that both interpolate and preserve the monotonicity of the input data, and we derive a simple ratio condition that guarantees the continuous differentiability of the limit function. We then show that the condition holds for both a scheme of Kuijt and van Damme, based on rational functions, and a scheme of Sabin and Dodgson, based on square roots. MSC: ...

Let (ΣA, σ) be a subshift of finite type and let M(x) be a continuous function on ΣA taking values in the set of non-negative matrices. We extend the classical scalar pressure function to this new setting and prove the existence of the Gibbs measure and the differentiability of the pressure function. We are especially interested on the case where M(x) takes finite values M1, · · · ,Mm. The pres...