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In this paper we analyze a dynamic recursive extension of the (static) notionof a deviation measure and its properties. We study distribution invariantdeviation measures and show that the only dynamic deviation measure which islaw invariant and recursive is the variance. We also solve the problem ofoptimal risk-sharing generalizing classical risk-sharing results for variancethrough a dynamic inf-convolution problem involving a transformation of theoriginal dynamic deviation measures.

Representation Results for Law Invariant Recursive Dynamic Deviation Measures and Risk Sharing