scholarly journals The strong Fatou property of risk measures

2018 ◽  
Vol 6 (1) ◽  
pp. 183-196 ◽  
Author(s):  
Shengzhong Chen ◽  
Niushan Gao ◽  
Foivos Xanthos
Keyword(s):  
Risk Measures ◽  
Lower Semicontinuous ◽  
Rearrangement Invariant Space ◽  
Invariant Space ◽  
Fatou Property ◽  
Rearrangement Invariant Spaces ◽  
Dual Representations ◽  
Wide Range ◽  
Invariant Functional ◽  
Invariant Spaces

AbstractIn this paper, we explore several Fatou-type properties of risk measures. The paper continues to reveal that the strong Fatou property,whichwas introduced in [19], seems to be most suitable to ensure nice dual representations of risk measures. Our main result asserts that every quasiconvex law-invariant functional on a rearrangement invariant space X with the strong Fatou property is (X, L1) lower semicontinuous and that the converse is true on a wide range of rearrangement invariant spaces. We also study inf-convolutions of law-invariant or surplus-invariant risk measures that preserve the (strong) Fatou property.

scholarly journals Diffeomorphic approximation of Planar Sobolev Homeomorphisms in rearrangement invariant spaces

2021 ◽  
Author(s):  
Daniel Campbell ◽  
Luigi Greco ◽  
Roberta Schiattarella ◽  
Filip Soudsky
Keyword(s):  
Rearrangement Invariant Space ◽  
Invariant Space ◽  
Rearrangement Invariant Spaces ◽  
Invariant Spaces

Let $\Omega\subseteq\mathcal{R}^2$ be a domain, let $X$ be a rearrangement invariant space and let $f\in W^{1}X(\Omega,\mathcal{R}^2)$ be a homeomorphism between $\Omega$ and $f(\Omega)$. Then there exists a sequence of diffeomorphisms $f_k$ converging to $f$ in the space $W^{1}X(\Omega,\mathcal{R}^2)$.

scholarly journals New classes of rearrangement-invariant spaces appearing in extreme cases of weak interpolation

2006 ◽  
Vol 4 (3) ◽  
pp. 275-304 ◽  
Author(s):  
Evgeniy Pustylnik ◽  
Teresa Signes
Keyword(s):  
Weak Type ◽  
Interpolation Theorem ◽  
Optimal Interpolation ◽  
Rearrangement Invariant Space ◽  
Invariant Space ◽  
Type Interpolation ◽  
Rearrangement Invariant Spaces ◽  
Invariant Spaces ◽  
Quasiconcave Function

We study weak type interpolation for ultrasymmetric spacesL?,Ei.e., having the norm??(t)f*(t)?E˜, where?(t)is any quasiconcave function andE˜is arbitrary rearrangement-invariant space with respect to the measuredt/t. When spacesL?,Eare not “too close” to the endpoint spaces of interpolation (in the sense of Boyd), the optimal interpolation theorem was stated in [13]. The case of “too close” spaces was studied in [15] with results which are optimal, but only among ultrasymmetric spaces. In this paper we find better interpolation results, involving new types of rearrangement-invariant spaces,A?,b,EandB?,b,E, which are described and investigated in detail.

scholarly journals Embeddings of homogeneous Sobolev spaces on the entire space

2020 ◽  
pp. 1-33 ◽  
Author(s):  
Zdeněk Mihula
Keyword(s):  
Sobolev Spaces ◽  
Orlicz Spaces ◽  
Function Spaces ◽  
The Other ◽  
Rearrangement Invariant Space ◽  
Entire Space ◽  
Invariant Space ◽  
One Dimensional ◽  
Homogeneous Sobolev Spaces ◽  
Invariant Spaces

Abstract We completely characterize the validity of the inequality $\| u \|_{Y(\mathbb R)} \leq C \| \nabla^{m} u \|_{X(\mathbb R)}$ , where X and Y are rearrangement-invariant spaces, by reducing it to a considerably simpler one-dimensional inequality. Furthermore, we fully describe the optimal rearrangement-invariant space on either side of the inequality when the space on the other side is fixed. We also solve the same problem within the environment in which the competing spaces are Orlicz spaces. A variety of examples involving customary function spaces suitable for applications is also provided.

scholarly journals INF-Convolution of Risk Measures on Rearrangement Invariant Spaces

2021 ◽  
Author(s):  
Shengzhong Chen
Keyword(s):  
Risk Measures ◽  
Model Space ◽  
Continuity Condition ◽  
Individual Risk ◽  
Risk Allocation ◽  
Fatou Property ◽  
Economic Agents ◽  
Research Areas ◽  
Optimal Capital ◽  
Invariant Spaces

The problem of optimal capital and risk allocation among economic agents, has played a predominant role in the respective academic and industrial research areas for decades. Typically as risk occurs in face of randomness the risks which are to be measured are identified with real-valued random variables on some probability space (Ω, F, P). Consider a model space X , and n economic agents with initial endowments X1, · · · , Xn ∈ X who assess the riskiness of their positions by means of law-invariant convex risk measures ρi : X → (−∞,∞]. In order to minimize total and individual risk, the agents redistribute the aggregate endowment X = X1 + · · · + Xn among themselves. An optimal capital and risk allocation Y1, · · · , Yn satisfies Y1 + · · · + Yn = X and ρ1(Y1) + · · · + ρ(Yn) = inf nXn i=1 ρi(Xi) : Xi ∈ X , i = 1, . . . , n, and Xn i=1 Xi = X o , (0.1) where n i=1ρi(X) = inf nPn i=1 ρi(Xi) : Xi ∈ X , i = 1, . . . , n, and Pn i=1 Xi = X o is the inf-convolution of ρ1, ..., ρn. In 2008, Filipovi´c and Svindland proved that if X is an L p (P) for some 1 ≤ p ≤ ∞ and ρi satisfy a suitable continuity condition (i.e. Fatou property), then Problem (0.1) always admits a solution. To reflect the fact of randomness of risk, we should consider the model space X chosen for risk evaluations to be as general as possible. The main contribution of this thesis is Theorem 4.10 has been published in [9]. It extends Filipovi´c and Svindland’s result from L p spaces to general rearrangement invariant (r.i.) spaces.

scholarly journals INF-Convolution of Risk Measures on Rearrangement Invariant Spaces

2021 ◽  
Author(s):  
Shengzhong Chen
Keyword(s):  
Risk Measures ◽  
Model Space ◽  
Continuity Condition ◽  
Individual Risk ◽  
Risk Allocation ◽  
Fatou Property ◽  
Economic Agents ◽  
Research Areas ◽  
Optimal Capital ◽  
Invariant Spaces

The problem of optimal capital and risk allocation among economic agents, has played a predominant role in the respective academic and industrial research areas for decades. Typically as risk occurs in face of randomness the risks which are to be measured are identified with real-valued random variables on some probability space (Ω, F, P). Consider a model space X , and n economic agents with initial endowments X1, · · · , Xn ∈ X who assess the riskiness of their positions by means of law-invariant convex risk measures ρi : X → (−∞,∞]. In order to minimize total and individual risk, the agents redistribute the aggregate endowment X = X1 + · · · + Xn among themselves. An optimal capital and risk allocation Y1, · · · , Yn satisfies Y1 + · · · + Yn = X and ρ1(Y1) + · · · + ρ(Yn) = inf nXn i=1 ρi(Xi) : Xi ∈ X , i = 1, . . . , n, and Xn i=1 Xi = X o , (0.1) where n i=1ρi(X) = inf nPn i=1 ρi(Xi) : Xi ∈ X , i = 1, . . . , n, and Pn i=1 Xi = X o is the inf-convolution of ρ1, ..., ρn. In 2008, Filipovi´c and Svindland proved that if X is an L p (P) for some 1 ≤ p ≤ ∞ and ρi satisfy a suitable continuity condition (i.e. Fatou property), then Problem (0.1) always admits a solution. To reflect the fact of randomness of risk, we should consider the model space X chosen for risk evaluations to be as general as possible. The main contribution of this thesis is Theorem 4.10 has been published in [9]. It extends Filipovi´c and Svindland’s result from L p spaces to general rearrangement invariant (r.i.) spaces.

scholarly journals On the extension property of dilatation monotone risk measures

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Massoomeh Rahsepar ◽  
Foivos Xanthos
Keyword(s):  
Risk Measures ◽  
Lower Semicontinuous ◽  
Natural Extension ◽  
Random Variables ◽  
Extension Property ◽  
Fatou Property ◽  
Quasi Convexity

Abstract Let 𝒳 be a subset of L 1 L^{1} that contains the space of simple random variables ℒ and ρ : X → ( - ∞ , ∞ ] \rho\colon\mathcal{X}\to(-\infty,\infty] a dilatation monotone functional with the Fatou property. In this note, we show that 𝜌 extends uniquely to a σ ⁢ ( L 1 , L ) \sigma(L^{1},\mathcal{L}) lower semicontinuous and dilatation monotone functional ρ ¯ : L 1 → ( - ∞ , ∞ ] \overline{\rho}\colon L^{1}\to(-\infty,\infty] . Moreover, ρ ¯ \overline{\rho} preserves monotonicity, (quasi)convexity and cash-additivity of 𝜌. We also study conditions under which ρ ¯ \overline{\rho} preserves finiteness on a larger domain. Our findings complement extension and continuity results for (quasi)convex law-invariant functionals. As an application of our results, we show that transformed norm risk measures on Orlicz hearts admit a natural extension to L 1 L^{1} that retains robust representations.

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