## The Annals of Probability

### Representations of Markov Processes as Multiparameter Time Changes

Thomas G. Kurtz

#### Abstract

Let $Y_1, Y_2,\cdots$ be independent Markov processes. Solutions of equations of the form $Z_i(t) = Y_i(\int^t_0\beta_i(Z(s))ds)$, where $\beta_i(z) \geqslant 0$, are considered. In particular it is shown that, under certain conditions, the solution of this "random time change problem" is equivalent to the solution of a corresponding martingale problem. These results give representations of a large class of diffusion processes as solutions of $X(t) = X(0) + \sum^N_{i=1}\alpha_iW_i(\int^t_0\beta_i(X(s))ds)$ where $\alpha_i \in \mathbb{R}^d$ and the $W_i$ are independent Brownian motions. A converse to a theorem of Knight on multiple time changes of continuous martingales is given, as well as a proof (along the lines of Holley and Stroock) of Liggett's existence and uniqueness theorems for infinite particle systems.

#### Article information

Source
Ann. Probab., Volume 8, Number 4 (1980), 682-715.

Dates
First available in Project Euclid: 19 April 2007

https://projecteuclid.org/euclid.aop/1176994660

Digital Object Identifier
doi:10.1214/aop/1176994660

Mathematical Reviews number (MathSciNet)
MR577310

Zentralblatt MATH identifier
0442.60072

JSTOR