On connection between random curves, changes of time and regenerative times of random processes

B. P. Harlamov

Zapiski Nauchnykh Seminarov POMI, Problems of the theory of probability distributions. Part 3, Tome 55 (1976), pp. 128-164

Résumé

A product space $\Phi\times D$ is considered where $\Phi$ is a set of all continuous non-decreasing functions $\varphi\colon[0,\infty)\to(0,\infty)$, $\varphi(0)=0$, $\varphi(t)\to+\infty$ ($t\to\infty$); $D$ is a set of all right-continuous functions $\xi\colon(0,\infty)\to X$, $X$ is some metric space. Two maps $\Phi\times D\to D$: are defined. The first is the projection $q(\varphi,\xi)=\xi$, and the second is change of time $u(\varphi,\xi)=\xi\circ\varphi$. The following equivalence relation in $D$ is defined: $$ \zeta_1\sim\xi_2\Leftrightarrow\exists\varphi_1,\varphi_2\in\Phi: \xi_1\circ\varphi_1=\xi_2\circ\varphi_2. $$ Let $M$ is a set of all equivalence classes. Then $L$ is the map $D\to M$: $L\xi_1=L\xi_2\Leftrightarrow\xi_1\sim\xi_2$. $L\xi$ is called the curve corresponding to $\xi$. The following theorem is proved: two random processes with probability measures $P^1$ and $P^2$ on $D$ possess of identical random curves (i.e. $P^1\circ L^{-1}=P^2\circ L^{-1}$) if and only if two random changes of time exist (i.e. two probability measures $Q^1$ and $Q^2$ on $\Phi\times D$) for which $P^1=Q^1\circ q^{-1}$, $P^2=Q^2\circ q^{-1}$) which transform these two processes in a process with a measure $\widetilde{P}$ (i.e. $Q^1\circ u^{-1}=Q^2\circ u^{-1}=\widetilde{P}$). If $(P_x^1)_{x\in X}$ and $(P_x^2)_{x\in X}$ are two families of probability measures for which $P_x^1\circ L^{-1}=P_x^2\circ L^{-1}$ $\forall x\in X$ then for each $x\in X$ corresponding measures $Q^1_x$ and $Q^2_x$ may be found as follows. The set of regenerative times of the family $(\widetilde{P}_x)_{x\in X}$ contains all stopping times which are simultaneously regenerative times of the families $(P^1_x)_{x\in X}$ and $(P^2_x)_{x\in X}$ and have a special first passage time property.