Hazard rate and reversed hazard rate monotonicities in continuous-time Markov chains

Abstract

A continuous-time Markov chain on the non-negative integers is called skip-free to the right (left) if only unit increments to the right (left) are permitted. If a Markov chain is skip-free both to the right and to the left, it is called a birth–death process. Karlin and McGregor (1959) showed that if a continuous-time Markov chain is monotone in the sense of likelihood ratio ordering then it must be an (extended) birth–death process. This paper proves that if an irreducible Markov chain in continuous time is monotone in the sense of hazard rate (reversed hazard rate) ordering then it must be skip-free to the right (left). A birth–death process is then characterized as a continuous-time Markov chain that is monotone in the sense of both hazard rate and reversed hazard rate orderings. As an application, the first-passage-time distributions of such Markov chains are also studied.

Monotonicity and Convexity of Option Prices Revisited

“…Recall that in Lemma 2.1 the transition probability density q, (x, y ) is TP2 in x and y if and only if the process is a diffusion. According to Kijima (1998a), it is possible to construct a stock price process that has jumps, but whose survival transition probability Q, (x,y) is TP2 in x and y. Hence, for such a stock price process, the monotonicity result (proved in Proposition 3.1 below) still holds true.…”

Towards Practical and Synthetical Modelling of Repairable Systems

“…A virtual age model transforms in essence the repair improvement into system age recovery (or shifts the age of the system to the left along the time axis). Combining such an age recovery idea and the local Poisson property of the general point processes Kijima (1989Kijima ( , 1998, Stadje and Zuckerman (1996), Finkelstain (1993, 1998), Makis and Jardine (1993), Guo (1993, 1996) However, a serious question arises in practice: How large is the improvement by repairs statistically? Our simulation results show that even with one percent improvement at each repair, the system will seriously move away from its baseline intensity function.…”

“…Non-linear dynamics may cause a chaotic behavior here. Therefore, explorations of general properties of failure/repair processes have gained more attention in recent years, for example, Boland and El-Neweihi (1998), Dagpunar (1998), Gong, Pruett, Tang (1997), Kijima (1998), Li, Shi and Cao (1997), Love and Guo (1994), Guo and Love (1994), and Stadje and Zuckerman (1996) At this point the fact should be acknowledged that the most important development in virtual age model research is due to Last and Szekli (1998), who successfully developed a general form of an age model based on the dynamics of marked point processes (Last and Brandt (1995)). …”

Testing behavior of the reversed hazard rate

“…doi:10.1016/j.apm.2010.11.054 of RHR. Kijima [7] has shown that certain first-passage times, associated with some continuous time Markov chains, has IHR if the underlying chain is monotone in the sense of the RHR order. Shanthikumar et al [8] have shown that if the service times of servers in a tandem queue with blocking are comparable in the RHR order, then there exists an optimal allocation where the server allocated to first stage has a larger mean service time than that assigned to the second server.…”