Comparison of level-crossing times for Markov and semi-Markov processes

We extend the definition of level-crossing ordering of stochastic processes, proposed by Irle and Gani (2001), to the case in which the times to exceed levels are compared using an arbitrary stochastic order, and work, in particular, with integral stochastic orders closed for convolution. Using a sample-path approach, we establish level-crossing ordering results for the case in which the slower of the processes involved in the comparison is skip-free to the right. These results are specially useful in simulating processes that are ordered in level crossing, and extend results of Irle and Gani (2001), Irle (2003), and Ferreira and Pacheco (2005) for skip-free-to-the-right discrete-time Markov chains, semi-Markov processes, and continuous-time Markov chains, respectively.

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Stochastic ordering for continuous-time processes

Journal of Applied Probability ◽

10.1239/jap/1053003549 ◽

2003 ◽

Vol 40 (2) ◽

pp. 361-375 ◽

Cited By ~ 6

Author(s):

A. Irle

Keyword(s):

Markov Chains ◽

Markov Processes ◽

Discrete Time ◽

Continuous Time ◽

Stochastic Ordering ◽

Level Crossing ◽

Birth And Death Processes ◽

Wiener Processes ◽

Continuous Time Processes ◽

Semi Markov Processes

We consider the following ordering for stochastic processes as introduced by Irle and Gani (2001). A process (Yt)t is said to be slower in level crossing than a process (Zt)t if it takes (Yt)t stochastically longer than (Zt)t to exceed any given level. In Irle and Gani (2001), this ordering was investigated for Markov chains in discrete time. Here these results are carried over to semi-Markov processes with particular attention to birth-and-death processes and also to Wiener processes.

Download Full-text

Stochastic ordering for continuous-time processes

Journal of Applied Probability ◽

10.1017/s0021900200019355 ◽

2003 ◽

Vol 40 (02) ◽

pp. 361-375 ◽

Cited By ~ 5

Author(s):

A. Irle

Keyword(s):

Markov Chains ◽

Markov Processes ◽

Discrete Time ◽

Continuous Time ◽

Stochastic Ordering ◽

Level Crossing ◽

Birth And Death Processes ◽

Wiener Processes ◽

Continuous Time Processes ◽

Semi Markov Processes

We consider the following ordering for stochastic processes as introduced by Irle and Gani (2001). A process (Y t ) t is said to be slower in level crossing than a process (Z t ) t if it takes (Y t ) t stochastically longer than (Z t ) t to exceed any given level. In Irle and Gani (2001), this ordering was investigated for Markov chains in discrete time. Here these results are carried over to semi-Markov processes with particular attention to birth-and-death processes and also to Wiener processes.

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Level-crossing ordering of semi-Markov processes and markov chains

Journal of Applied Probability ◽

10.1239/jap/1134587811 ◽

2005 ◽

Vol 42 (4) ◽

pp. 989-1002 ◽

Cited By ~ 2

Author(s):

Fátima Ferreira ◽

António Pacheco

Keyword(s):

Markov Chains ◽

Markov Processes ◽

Stochastic Order ◽

Sample Path ◽

Stochastic Orders ◽

Level Crossing ◽

The Times ◽

Definition Of ◽

The Right ◽

Semi Markov Processes

We extend the definition of level-crossing ordering of stochastic processes, proposed by Irle and Gani (2001), to the case in which the times to exceed levels are compared using an arbitrary stochastic order, and work, in particular, with integral stochastic orders closed for convolution. Using a sample-path approach, we establish level-crossing ordering results for the case in which the slower of the processes involved in the comparison is skip-free to the right. These results are specially useful in simulating processes that are ordered in level crossing, and extend results of Irle and Gani (2001), Irle (2003), and Ferreira and Pacheco (2005) for skip-free-to-the-right discrete-time Markov chains, semi-Markov processes, and continuous-time Markov chains, respectively.

Download Full-text