Two-dimensional π - Comonotonic Additive Risk Measure

The theory of risk measure is such a kind of method that using certain kind of statistical magnitude of random variables which describe the random loses of certain events to evaluate and quantify risks.And by the same time,based on these statistical magnitudes,we can take good care of risks contained in everyday economic activities.Also,we can give advices about how to properly allocate capital.Since nowadays,commonly used one-period risk measure models include Value-at-Risk,coherent risk measure,convex risk measure and distortion risk measure.While these models talk less about what happens towards all the significants of a certain portfolio,some models based on multivariate risks have been raised up these days.In the first part of our article,we take a review about these one-dimensional models,and introduce to the readers theorems about scalar-valued,set-valued and vector-valued multivariate risk measures.While in part two,we use several chapters generalizing the theorem of distortion risk measure which was raised up by Wang in 1997 to multidimensional situation,and then establish our vector-valued risk measure based on copula.Then we just show how our theorem connects with the existing theorems,and give some discussions about the classical properties that a ’good’ risk measure should hold.