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.
Publié le : 1980-08-14
Classification:
Markov processes,
diffusion processes,
martingale problem,
random time change,
multiparameter martingales,
infinite particle systems,
stopping times,
continuous martingales,
60J25,
60G40,
60G45,
60J60,
60J75,
60K35
@article{1176994660,
author = {Kurtz, Thomas G.},
title = {Representations of Markov Processes as Multiparameter Time Changes},
journal = {Ann. Probab.},
volume = {8},
number = {6},
year = {1980},
pages = { 682-715},
language = {en},
url = {http://dml.mathdoc.fr/item/1176994660}
}
Kurtz, Thomas G. Representations of Markov Processes as Multiparameter Time Changes. Ann. Probab., Tome 8 (1980) no. 6, pp. 682-715. http://gdmltest.u-ga.fr/item/1176994660/