The It\^{o} integral with respect to an infinite dimensional L\'{e}vy process: A series approach

We present an alternative construction of the infinite dimensional It^{o} integral with respect to a Hilbert space valued L'{e}vy process. This approach is based on the well-known theory of real-valued stochastic integration, and the respective It^{o} integral is given by a series of It^{o} integrals with respect to standard L'{e}vy processes. We also prove that this stochastic integral coincides with the It^{o} integral that has been developed in the literature.