This paper studies multidimensional dynamic risk measure induced by conditional g-expectation . 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-expectation and multidimensional dynamic risk measure ; 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 also given for multidimensional dynamic convex g-risk measure in which the penalty term is expressed more precisely. Several examples are presented to show how this multidimensional approach is applied to a class of agent-based model and to the problem of risk allocation .
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 .
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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. As to applications, we
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