Gaussian quasi-Markov processes with stationary increments. (Russian) Zbl 0646.60045

Molchan, G. M.

Teor. Veroyatn. Mat. Stat., Kiev 37, 105-111 (1987).

Let \(x(t),\) \(0\leq t\leq T\), be a continuous in mean Gaussian process with stationary increments and \[ E| x(t)-x(s)|^ 2=\Delta (| t- s|). \] It is proved that \(x(t)\) is a quasi-Markov process if and only if \[ \Delta(t) = c^ 2 \cdot \begin{cases} 1-\exp(-\kappa t)/\kappa, & \kappa\geq 0, T\leq \infty, \\ \kappa t+t^ 2, & \kappa>0, T\leq\infty, \\ [\cosh \kappa L-\cosh\kappa(L-t)] / \kappa^ 2, & \kappa\leq 0, T\leq 2L<\infty, \\ [\sinh\kappa L-\sinh\kappa(L-t)] / \kappa, & \kappa >0, T\leq 2L<\infty, \\ [\sin\kappa L-\sin\kappa(L-t)] / \kappa, & 0<2L\kappa<\pi, T\leq 2L \\ [1-\cos\kappa t] / \kappa^ 2, & \kappa\geq 0, 0\leq\kappa T\leq\pi, \end{cases} \] for \(\kappa =0\), \(\Delta(t)=\lim \Delta(\kappa t)\), \(\kappa\to 0\).

Reviewer: A.Plikusas

MSC:

60G15	Gaussian processes
60J27	Continuous-time Markov processes on discrete state spaces

Keywords:

Gaussian process with stationary increments; quasi-Markov process