Abstract:

The notion of a process $X(t)$ with independent increments is generalized. It is required that for $0=t_0<t_1<\cdots <t_n\leq T $ the r. v. $ X(t_0), X(t_1)-\alpha (t_0,t_1) X(t_0),\ldots , X(t_n)-\alpha (t_{n-1},t_n) X(t_{n-1})$ are independent with some suitable function $\alpha(s,t)$. This class consists of Markov processes with a special structure of transition kernels and includes both the processes with independent increments and the regular Markov processes introduced by Vajda. The main result is that under some mild additional conditions every continuous process from this class is a Gauss-Markov-process.