Abstract In this paper, we study capital allocation for dynamic risk measures, with an axiomatic approach but also by exploiting the relation between measures and BSDEs. Although there is a wide literature on rules in static setting only few recent papers work and, moreover, those mainly focus gradient approach. To fill gap, then discuss new perspectives to problem going beyond already existing...

It is known that directional differentiability of metric projection onto a closed convex set in a finite dimensional space is not guaranteed. In this paper we discuss sufficient conditions ensuring directional differentiability of such metric projections. The approach is based on a general theory of sensitivity analysis of parameterized optimization problems.

Peano differentiability is a notion of higher order differentiability in the ordinary sense. H. W. Oliver gave sufficient conditions for the mth Peano derivative to be a derivative in the ordinary sense in the case of functions of a real variable. Here we generalize this theorem to functions of several variables.

We propose a direct method to control the first order fractional difference quotients of solutions to quasilinear subelliptic equations in the Heisenberg group. In this way we implement iteration methods on fractional difference quotients to obtain weak differentiability in the T -direction and then second order weak differentiability in the horizontal directions.

In this paper, we show that the metric space (Q,G) is a positivelycurved space (PC-space) in the sense of Alexandrov. We also discuss some issues like metric tangent cone and exponential map of (Q, G). Then we give a stratification of this metric space according to the signature of points in Q. Some properties of this stratification are shown. The second part of this paper is devoted to some ba...

In this paper, we show that the metric space (Q,G) is a positivelycurved space (PC-space) in the sense of Alexandrov. We also discuss some issues like metric tangent cone and exponential map of (Q, G). Then we give a decomposition of this metric space according to the signature of points in Q. Some properties of this decomposition are shown. The second part of this paper is devoted to some basi...

We address the question of whether geometric conditions on given data can be preserved by a solution in (1) Whitney extension problem, and (2) Brenner-Fefferman-Hochster-Koll\'ar both for $\mathcal C^m$ functions. Our results involve certain loss differentiability. Problem concerns system linear equations $A(x)G(x)=F(x)$, where $A$ is matrix functions $\mathbb R^n$, $F$, $G$ are vector-valued S...