Stochastic PDEs in S′ for SDEs driven by Levy noise

In this article we show that a finite-dimensional stochastic differential equation driven by a Levy noise can be formulated as a stochastic partial differential equation (SPDE) driven by the same Levy noise. We prove the existence result for such an SPDE by Ito's formula for translation operators, and the uniqueness by an adapted form of "Monotonicity inequality", proved earlier in the diffusion case. As a consequence, the solutions that we construct have the "translation invariance" property.