On stationary solutions of delay differential equations driven by a Lévy process

The stochastic delay differential equationdX(t)=∫[−r,0]X(t+u) a(du) dt+dZ(t), t⩾0is considered, where Z(t) is a process with independent stationary increments and a is a finite signed measure. We obtain necessary and sufficient conditions for the existence of a stationary solution to this equation in terms of a and the Lévy measure of Z.