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August, 1980 Representations of Markov Processes as Multiparameter Time Changes
Thomas G. Kurtz
Ann. Probab. 8(4): 682-715 (August, 1980). DOI: 10.1214/aop/1176994660

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.

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Thomas G. Kurtz. "Representations of Markov Processes as Multiparameter Time Changes." Ann. Probab. 8 (4) 682 - 715, August, 1980. https://doi.org/10.1214/aop/1176994660

Information

Published: August, 1980
First available in Project Euclid: 19 April 2007

zbMATH: 0442.60072
MathSciNet: MR577310
Digital Object Identifier: 10.1214/aop/1176994660

Subjects:
Primary: 60J25
Secondary: 60G40, 60G45, 60J60, 60J75, 60K35

Rights: Copyright © 1980 Institute of Mathematical Statistics

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Vol.8 • No. 4 • August, 1980
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