# Approximation of bivariate Markov chains by one-dimensional diffusion processes

Aplikace matematiky (1978)

• Volume: 23, Issue: 4, page 267-279
• ISSN: 0862-7940

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## Abstract

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The paper deals with several questions of the diffusion approximation. The goal of this paper is to create the general method of reducting the dimension of the model with the aid of the diffusion approximation. Especially, two dimensional random variables are approximated by one-dimensional diffusion process by replacing one of its coordinates by a certain characteristic, e.g. by its stationary expectation. The suggested method is used for several different systems. For instance, the method is applicable to the sequences of Markov chains $\left\{{\left(}^{n}{X}_{m}{,}^{n}{Y}_{m}\right),m=0,1,...\right\}\phantom{\rule{4pt}{0ex}}n=1,2,...$ where the tendency of $\left\{{}^{n}{Y}_{m}\right\}$ to the stationary state is greater than that of $\left\{{}^{n}{X}_{m}\right\}$.

## How to cite

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Kuklíková, Daniela. "Approximation of bivariate Markov chains by one-dimensional diffusion processes." Aplikace matematiky 23.4 (1978): 267-279. <http://eudml.org/doc/15056>.

@article{Kuklíková1978,
abstract = {The paper deals with several questions of the diffusion approximation. The goal of this paper is to create the general method of reducting the dimension of the model with the aid of the diffusion approximation. Especially, two dimensional random variables are approximated by one-dimensional diffusion process by replacing one of its coordinates by a certain characteristic, e.g. by its stationary expectation. The suggested method is used for several different systems. For instance, the method is applicable to the sequences of Markov chains $\left\lbrace (^nX_m, ^nY_m), m=0,1,\ldots \right\rbrace \ n=1,2,\ldots$ where the tendency of $\left\lbrace ^nY_m \right\rbrace$ to the stationary state is greater than that of $\left\lbrace ^nX_m\right\rbrace$.},
author = {Kuklíková, Daniela},
journal = {Aplikace matematiky},
keywords = {diffusion approximation; Markov chains; ito equation; difference equation; renewal process; Diffusion Approximation; Markov Chains; Ito Equation; Difference Equation; Renewal Process},
language = {eng},
number = {4},
pages = {267-279},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Approximation of bivariate Markov chains by one-dimensional diffusion processes},
url = {http://eudml.org/doc/15056},
volume = {23},
year = {1978},
}

TY - JOUR
AU - Kuklíková, Daniela
TI - Approximation of bivariate Markov chains by one-dimensional diffusion processes
JO - Aplikace matematiky
PY - 1978
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 23
IS - 4
SP - 267
EP - 279
AB - The paper deals with several questions of the diffusion approximation. The goal of this paper is to create the general method of reducting the dimension of the model with the aid of the diffusion approximation. Especially, two dimensional random variables are approximated by one-dimensional diffusion process by replacing one of its coordinates by a certain characteristic, e.g. by its stationary expectation. The suggested method is used for several different systems. For instance, the method is applicable to the sequences of Markov chains $\left\lbrace (^nX_m, ^nY_m), m=0,1,\ldots \right\rbrace \ n=1,2,\ldots$ where the tendency of $\left\lbrace ^nY_m \right\rbrace$ to the stationary state is greater than that of $\left\lbrace ^nX_m\right\rbrace$.
LA - eng
KW - diffusion approximation; Markov chains; ito equation; difference equation; renewal process; Diffusion Approximation; Markov Chains; Ito Equation; Difference Equation; Renewal Process
UR - http://eudml.org/doc/15056
ER -

## References

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1. I. I. Gichman A. V. Skorochod, Theory of Random Processes III, (in Russian). Nauka, Moscow 1975. (1975) MR0651014
2. P. Mandl, A connection between Controlled Markov Chains and Martingales, Kybernetika 9 (1973), 237-241. (1973) Zbl0265.60060MR0323427
3. P. Mandl, On aggregating controlled Markov chains, in Jaroslav Hájek Memorial Volume (to appear). Zbl0431.93063MR0561266
4. P. Morton, 10.2307/3212178, J. Appl. Probability 8 (1971), 551 - 560. (1971) Zbl0234.49019MR0292572DOI10.2307/3212178

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