In this paper,we give an overview of representation theorems for various static risk measures:coherent or convex risk measures, risk measures with comonotonic subadditivity or convexity, law-invariant coherent or convex risk measures, risk measures with comonotonic subadditivity or convexity and respecting stochastic orders.
This article gives the representations of two types of real functionals on Z∞(Ω,F) or L∞(Ω,F,P) in terms of Choquet integrals. These functionals are comonotonically subadditive and comonotonically convex, respectively.
This paper surveys some results on Wick product and Wick renormalization. The framework is the abstract Wiener space. Some known results on Wick product and Wick renormalization in the white noise analysis framework are presented for classical random variables. Some conditions are described for random variables whose Wick product or whose renormalization are integrable random variables. Relevant results on multiple Wiener integrals, second quantization operator, Malliavin calculus and their relations with the Wick product and Wick renormalization are also briefly presented. A useful tool for Wick product is the S-transform which is also described without the introduction of generalized random variables.
Continuous-time Markowitz＇s by parameterizing a critical quantity. It mean-variance efficient strategies are modified is shown that these parameterized Markowitz strategies could reach the original mean target with arbitrarily high probabilities. This, in turn, motivates the introduction of certain stopped strategies where stock holdings are liquidated whenever the parameterized Markowitz strategies reach the present value of the mean target. The risk aspect of the revised Markowitz strategies are examined via expected discounted loss from the initial budget. A new portfolio selection model is suggested based on the results of the paper.
Considering the classical model with risky investment, we are interested in the ruin probability that is minimized by a suitably chosen investment strategy for a capital market index. For claim sizes with common distribution of extended regular variation, starting from an integro-differential equation for the maximal survival probability, we find that the corresponding ruin probability as a function of the initial surplus is also extended regular variation.
