This paper deals with multidimensional dynamic risk measures induced by conditional g-expectations. A notion of multidimensional g-expectation is proposed to provide a multidimensional version of nonlinear expectations. By a technical result on explicit expressions for the comparison theorem, uniqueness theorem and viability on a rectangle of solutions to multidimensional backward stochastic differential equations, some necessary and sufficient conditions are given for the constancy, monotonicity, positivity, homogeneity and translatability properties of multidimensional conditional g-expectations and multidimensional dynamic risk measures; we prove that a multidimensional dynamic g-risk measure is nonincreasingly convex if and only if the generator g satisfies a quasi-monotone increasingly convex condition. A general dual representation is given for the multidimensional dynamic convex g-risk measure in which the penalty term is expressed more precisely. Similarly to the one dimensional case, a sufficient condition for a multidimensional dynamic risk measure to be a g-expectation is also explored . As to applications, we show how this multidimensional approach can be applied to measure the insolvency risk of a firm with interacted subsidiaries \protect .
This paper deals with multidimensional dynamic risk measures induced by conditional g-expectations. A notion of multidimensional g-expectation is proposed to provide a multidimensional version of nonlinear expectations. By a technical result on explicit expressions for the comparison theorem, uniqueness theorem and viability on a rectangle of solutions to multidimensional backward stochastic differential equations, some necessary and sufficient conditions are given for the constancy, monotonicity, positivity, homogeneity and translatability properties of multidimensional conditional g-expectations and multidimensional dynamic risk measures; we prove that a multidimensional dynamic g-risk measure is nonincreasingly convex if and only if the generator g satisfies a quasi-monotone increasingly convex condition. A general dual representation is given for the multidimensional dynamic convex g-risk measure in which the penalty term is expressed more precisely. It is shown that model uncertainty leads to the convexity of risk measures . As to applications, we show how this multidimensional approach can be applied to measure the insolvency risk of a firm with interacted subsidiaries ; optimal risk sharing for\protect\gamma -tolerant g-risk measures is investigated. Insurance g-risk measure and other ways to induce g-risk measures are also studied at the end of the paper .
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Similarly to the one dimensional case, a sufficient condition for a multidimensional dynamic risk measure to be a g-expectation is also explored
It is shown that model uncertainty leads to the convexity of risk measures
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; optimal risk sharing for
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\gamma -tolerant g-risk measures is investigated. Insurance g-risk measure and other ways to induce g-risk measures are also studied at the end of the paper