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For a general Multidimensional L\'{e}vy process (satisfying some momentconditions), we introduce the Multidimensional power jump processes and therelated Multidimensional Teugels martingales. Furthermore, we orthogonalize theMultidimensional Teugels martingales by applying Gram-Schmidt process. We givea chaotic representation for every square integral random variable in terms ofthese orthogonalized Multidimensional Teugels martingales. The predictablerepresentation with respect to the same set of Multidimensional orthogonalizedmartingales of square integrable random variables and of square integrablemartingales is an easy consequence of the chaotic representation.

Chaotic and Predictable Representations for Multidimensional L\'{e}vy Processes