On stationary solutions of delay differential equations driven by a Lévy process. (Q1877511)

Let \(a\) be a finite signed measure on \([-r,0]\), \(Z\) a Lévy process (that is a real process with independent stationary increments and càdlàg paths). A linear stochastic delay differential equation \[ X(t) = X(0) + \int ^ {t}_ {0}\int _ {[-r,0]} X(s+u)\,da(u) \,ds + Z(t), \quad t\geq 0, \tag{1} \] driven by \(Z\) is studied, only càdlàg solutions to (1) such that \(Z\) and \((X(t),\,-r\leq t\leq 0)\) are independent being considered. Set \(h(\lambda ) = \lambda - \int _ {[-r,0]}\exp (\lambda u)\,da(u)\) and \(v_ {0} = \sup \{\operatorname {Re} \lambda \mid \lambda \in \mathbb C,\;h(\lambda )=0\}\). Let the Lévy measure of jumps of the process \(Z\) be denoted by \(F\). It is shown that there exists a stationary solution to (1) if and only if \(v_ {0}<0\) and \(\int _ {| y| >1} \log | y| \,dF(y) < \infty \). If \(X\) is a stationary solution to (1), then \(X(t)\) equals in law to \(\int ^ \infty _ 0 x_ 0(t)\,dZ(t)\), where \(x_ 0\) is the fundamental solution of the deterministic counterpart (\(Z\equiv 0\)) to (1).