The Annals of Probability

Representations of Markov Processes as Multiparameter Time Changes

Thomas G. Kurtz

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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

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

First available in Project Euclid: 19 April 2007

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Zentralblatt MATH identifier


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]

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


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

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