Stochastic PDEs in ${\mathcal{S}}^{\mathrm{\prime }}$ for SDEs driven by L\'evy noise

In this article we show that a finite dimensional stochastic differential equation driven by a L'evy process can be formulated as a stochastic partial differential equation. We prove the existence and uniqueness of strong solutions of such stochastic PDEs. The solutions that we construct have the `translation invariance’ property. The special case of this correspondence for diffusion processes was proved in [Rajeev, Translation invariant diffusion in the space of tempered distributions, Indian J. Pure Appl. Math. 44 (2013), no.~2, 231–258].