## The Annals of Probability

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

### Representations of Markov Processes as Multiparameter Time Changes

#### 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

**Permanent link to this document**

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**

links.jstor.org

**Subjects**

Primary: 60J25: Continuous-time Markov processes on general state spaces

Secondary: 60G40: Stopping times; optimal stopping problems; gambling theory [See also 62L15, 91A60] 60G45 60J60: Diffusion processes [See also 58J65] 60J75: Jump processes 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

**Keywords**

Markov processes diffusion processes martingale problem random time change multiparameter martingales infinite particle systems stopping times continuous martingales

#### Citation

Kurtz, Thomas G. Representations of Markov Processes as Multiparameter Time Changes. Ann. Probab. 8 (1980), no. 4, 682--715. doi:10.1214/aop/1176994660. https://projecteuclid.org/euclid.aop/1176994660