Efficient hedging with coherent risk measure

The idea of efficient hedging has been introduced by Föllmer and Leukert. They defined the shortfall risk as the expectation of the shortfall weighted by a loss function, and looked for strategies that minimize the shortfall risk under a capital constraint. In this paper, to measure the shortfall risk, we use the coherent risk measures introduced by Artzner, Delbaen, Eber and Heath. We show that, for a given contingent claim H, the optimal strategy consists in hedging a modified claim ?H for some randomized test ?. This is an analogue of the results by Föllmer and Leukert.     

Multivariate tail conditional expectation for elliptical distributions

In this paper we introduce a novel type of a multivariate tail conditional expectation (MTCE) risk measure and explore its properties. We derive an explicit closed-form expression for this risk measure for the elliptical family of distributions taking into account its variance–covariance dependency structure. As a special case we consider the normal, Student-t and Laplace distributions, important and popular in actuarial science and finance. The motivation behind taking the multivariate TCE for the elliptical family comes from the fact that unlike the traditional tail conditional expectation, the MTCE measure takes into account the covariation between dependent risks, which is the case when we are dealing with real data of losses. We illustrate our results using numerical examples in the case of normal and Student-t distributions.     

Biprobability logic with conditional expectation

This paper is devoted to fill the gap in studying logics for biprobability structures. We introduce the logic $L_{\mathbb AE_1E_2}^a$ with two conditional expectation operators and prove the completeness theorem. © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim     

Characterization of set‐valued conditional expectation

For an essentially bounded closed‐convex‐nonempty set‐valued random variable, it is shown that the conditional expectation is characterized by its integrals over the sets of the associated σ ‐field. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Computing the variance of a conditional expectation via non-nested Monte Carlo

Computing the variance of a conditional expectation has often been of importance in uncertainty quantification. Sun et al. has introduced an unbiased nested Monte Carlo estimator, which they call $1\frac{1}{2}$-level simulation since the optimal inner-level sample size is bounded as the computational budget increases. In this letter, we construct unbiased non-nested Monte Carlo estimators based on the so-called pick-freeze scheme due to Sobol’. An extension of our approach to compute higher order moments of a conditional expectation is also discussed.     

Second order regular variation and conditional tail expectation of multiple risks

For the purpose of risk management, the study of tail behavior of multiple risks is more relevant than the study of their overall distributions. Asymptotic study assuming that each marginal risk goes to infinity is more mathematically tractable and has also uncovered some interesting performance of risk measures and relationships between risk measures by their first order approximations. However, the first order approximation is only a crude way to understand tail behavior of multiple risks, and especially for sub-extremal risks. In this paper, we conduct asymptotic analysis on conditional tail expectation (CTE) under the condition of second order regular variation (2RV). First, the closed-form second order approximation of CTE is obtained for the univariate case. Then CTE of the form E[X₁∣g(X₁,…,X_d)>t], as t→∞, is studied, where g is a loss aggregating function and (X₁,…,X_d)?(RT₁,…,RT_d) with R independent of (T₁,…,T_d) and the survivor function of R satisfying the condition of 2RV. Closed-form second order approximations of CTE for this multivariate form have been derived in terms of corresponding value at risk. For both the univariate and multivariate cases, we find that the first order approximation is affected by only the regular variation index −α of marginal survivor functions, while the second order approximation is influenced by both the parameters for first and second order regular variation, and the rate of convergence to the first order approximation is dominated by the second order parameter only. We have also shown that the 2RV condition and the assumptions for the multivariate form are satisfied by many parametric distribution families, and thus the closed-form approximations would be useful for applications. Those closed-form results extend the study of Zhu and Li (submitted for publication).     

Some conditional expectation identities for the multivariate normal

We give formulas for the conditional expectations of a product of multivariate Hermite polynomials with multivariate normal arguments. These results are extended to include conditional expectations of a product of linear combination of multivariate normals. A unified approach is given that covers both Hermite and modified Hermite polynomials, as well as polynomials associated with a matrix whose eigenvalues may be both positive and negative.     

Multivariate risk measures based on conditional expectation and systemic risk for Exponential Dispersion Models

Exponential dispersion models are well used and studied in quantitative risk management and actuarial science. One of the main interests is the risk measurement analysis of such models when facing extreme loss events. In this paper, we propose two multivariate risk measures based on conditional expectation and derive the explicit formulae for exponential dispersion models. In particular, our multivariate risk measures could facilitate a systemic risk measure with explicit expressions for exponential dispersion models subject to any pre-specified “systemic event.” We provide two numerical examples based on practical data to show the advantages of our approach in the context of exponential dispersion models.     

Multiplication conditional expectation type operators on Orlicz spaces

In this paper we consider a generalized conditional-type Hölder inequality and investigate some classical properties of multiplication conditional expectation type operators on Orlicz spaces.     

Characterization of probability distributions by conditional expectation of function of record statistics

In this paper, two general classes of distributions have been characterized through conditional expectation of power of difference of two record statistics. Further, some particular cases and examples are also discussed.     

On some methods of deriving geometric expectation

Formulas presented for the calculation of ∑ _j=1 ⁿ j^k (n, k ∈ N do not have a closed form; they are in the form of recursive or complex formulas. Here an attempt is made to present a simple formula in which it is only necessary to compute the numerical coefficients in a recursive form, and the coefficients in turn follow a simple pattern (almost similar to Pascal's Triangle). Although the pattern for calculating numerical coefficients based on forming a table is easy, non-recursive formulas are presented to determine the numerical coefficients.     

Estimating the tails of loss severity via conditional risk measures for the family of symmetric generalised hyperbolic distributions

This paper addresses one of the main challenges faced by insurance companies and risk management departments, namely, how to develop standardised framework for measuring risks of underlying portfolios and in particular, how to most reliably estimate loss severity distribution from historical data. This paper investigates tail conditional expectation (TCE) and tail variance premium (TVP) risk measures for the family of symmetric generalised hyperbolic (SGH) distributions. In contrast to a widely used Value-at-Risk (VaR) measure, TCE satisfies the requirement of the “coherent” risk measure taking into account the expected loss in the tail of the distribution while TVP incorporates variability in the tail, providing the most conservative estimator of risk. We examine various distributions from the class of SGH distributions, which turn out to fit well financial data returns and allow for explicit formulas for TCE and TVP risk measures. In parallel, we obtain asymptotic behaviour for TCE and TVP risk measures for large quantile levels. Furthermore, we extend our analysis to the multivariate framework, allowing multivariate distributions to model combinations of correlated risks, and demonstrate how TCE can be decomposed into individual components, representing contribution of individual risks to the aggregate portfolio risk.     

On semiregular conditional distributions

Let (,A,P) denote some probability space and some sub--algebra ofA. It is shown that there exists a semiregular versionQ (A),A, , of the conditional distributionP(A|), AA, i.e., Q (A), (AA fixed) is andAQ (A),AA ( fixed), is a probability charge satisfyingQ (N)=0, , for allP-zero setsN, if and only ifL ₁(,P|) has a lifting, which exists for any sub--algebra ofA ifL ₁(,A P) is separable. Separability ofL ₁(,A,P) implies also the existence of a strongly semiregular versionQ (A),A, , ofP(A|), A , i.e., Q (A), (AA fixed), is -measurable andAQ (A),A ( fixed), is a probability charge. Furthermore,P can be written as P ₁+(1–)P ₂, 01, whereP ₁ are probability measures onA such thatP ₁(A|),AA, has a semiregular version vanishing for anyP-zero setN andP ₂ is singular with respect to any probability measure onA of the type ofP ₁. In the case 0<<1 the probability measuresP _j ,j=1, 2, are uniquely determined. The decomposition can be carried over to the case, where the additional condition thatQ (N)=0 for all and anyP-zero setN is valid, is omitted respectively semiregularity is replaced by (i) strong semiregularity, or (ii) classical regularity. In the last mentioned case (ii) the decomposition is multiplicative.     

一类马氏样本下假设检验问题错误概率的估计

本文考虑样本不独立情形的统计推断问题,研究基于马氏样本的最优势检验,给出了此情形的Neyman-Pearson基本引理.当样本容易足够大时,利用大偏差原理,得到了Neyman-Pearson型检验所犯两类错误概率的精确估计,它推广了经典Neyman-Pearson基本引理的相关结果.     

集值条件期望的收敛定理

本文给出了集合序列弱极限的表示定理，得到了随机集列关于σ－域流的条件期望序列在弱收敛意义的Fatou型引理和控制收敛定理。     

On the use of conditional expectation in portfolio selection problems

Annals of Operations Research - In this paper, we examine the use of conditional expectation, either to reduce the dimensionality of large-scale portfolio problems or to propose alternative...     

On conditional edge-connectivity of graphs

1. IntroductionIn this paper, a graph G ~ (V,E) always means a simple graph (without loops andmultiple edges) with the vertex-set V and the edge-set E. We follow [1] for graph-theoreticalterllilnology and notation not defined here.It is well known that when the underlying topology of a computer interconnectionnetwork is modeled by a graph G, the edge-connectivity A(G) of G is an important measurefor fault-tolerance of the network. However, it has many deficiencies (see [2]). MotiVatedby t…     

On the impact of conditional expectation estimators in portfolio theory

In this paper, we investigate the implications for portfolio theory of using conditional expectation estimators. First, we focus on the approximation of the conditional expectation within large-scale portfolio selection problems. In this context, we propose a new consistent multivariate kernel estimator to approximate the conditional expectation and it optimizes the bandwidth selection of kernel-type estimators. Second, we deal with the portfolio selection problem from the point of view of different non-satiable investors, namely risk-averse and risk-seeker investors. In particular, using a well-known ordering classification, we first identify different definitions of returns based on the investors preferences. Finally, for each problem, we examine several admissible portfolio optimization problems applied to the US stock market. The proposed empirical analysis allows us to evaluate the impact of the conditional expectation estimators in portfolio theory.