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

Stochastic Processes and their Applications. 2000. Vol. 88. No. 2. P. 195–211.

Гущин А. А., Кюхлер У.

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)+∫ 0 t ∫ [-r,0] X(s+u)da(u)ds+Z(t),t≥0,(1)

driven by Z is studied, only càdlàg solutions to (1) such that Z and (X(t),-r≤t≤0) are independent being considered. Set h(λ)=λ-∫ [-r,0] exp(λu)da(u) and v 0 =sup{Reλ∣λ∈ℂ,h(λ)=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 ∫ |y|>1 log|y|dF(y)<∞. If X is a stationary solution to (1), then X(t) equals in law to ∫ 0 ∞ x 0 (t)dZ(t), where x 0 is the fundamental solution of the deterministic counterpart (Z≡0) to (1).

Ключевые слова: Lévy processes, generalized Ornstein-Uhlenbeck processes, stationary solutions, stochastic delay equations