Stochastic equations for stationary Gaussian processes. (English. Russian original) Zbl 0745.60062

Ryzhov, Yu. M.

Theory Probab. Math. Stat. 42, 153-159 (1991); translation from Teor. Veroyatn. Mat. Stat., Kiev 42, 128-134 (1990).

Let \(\gamma\) be a measure on \([0,\infty)\), \(\sigma\geq 0\), and \(\omega(\cdot)\) a Brownian motion process. The author gives conditions for the existence and uniqueness of a stationary solution to the equation \[ \xi(t)=\xi(0)+\int^ t_ 0 \int^ \infty_ 0 \xi(s- u)\gamma(du)+\sigma\omega(t). \]

Reviewer: A.Dale (Durban)

MSC:

60H10	Stochastic ordinary differential equations (aspects of stochastic analysis)
60G15	Gaussian processes
60J65	Brownian motion
60G57	Random measures
60G30	Continuity and singularity of induced measures

Keywords:

stochastic equation; Gaussian process; Brownian motion; stationary solution