scholarly journals Conditional Processes Induced by Birth and Death Processes

2010 ◽  
Vol 2010 ◽  
pp. 1-24
Author(s):  
Masaru Iizuka ◽  
Matsuyo Tomisaki
Keyword(s):  
Stochastic Processes ◽  
State Space ◽  
Boundary Point ◽  
Markov Property ◽  
The Other ◽  
Birth And Death Processes ◽  
Left Boundary ◽  
Finite State ◽  
The Right ◽  
Finite State Space

For birth and death processes with finite state space, we consider stochastic processes induced by conditioning on hitting the right boundary point before hitting the left boundary point. We call the induced stochastic processes the conditional processes. We show that the conditional processes are again birth and death processes when the right boundary point is absorbing. On the other hand, it is shown that the conditional processes do not have Markov property and they are not birth and death processes when the right boundary point is reflecting.

Quasi-stationary distributions of birth-and-death processes

1978 ◽  
Vol 10 (03) ◽  
pp. 570-586 ◽  
Author(s):  
James A. Cavender
Keyword(s):  
State Space ◽  
Stationary Distribution ◽  
Parameter Family ◽  
Death Process ◽  
Birth And Death Process ◽  
Stationary Distributions ◽  
Birth And Death Processes ◽  
Finite State ◽  
Finite State Space ◽  
Quasi Stationary Distribution

Letqn(t) be the conditioned probability of finding a birth-and-death process in statenat timet,given that absorption into state 0 has not occurred by then. A family {q1(t),q2(t), · · ·} that is constant in time is a quasi-stationary distribution. If any exist, the quasi-stationary distributions comprise a one-parameter family related to quasi-stationary distributions of finite state-space approximations to the process.

Quasi-stationary distributions of birth-and-death processes

1978 ◽  
Vol 10 (3) ◽  
pp. 570-586 ◽  
Author(s):  
James A. Cavender
Keyword(s):  
State Space ◽  
Stationary Distribution ◽  
Parameter Family ◽  
Death Process ◽  
Birth And Death Process ◽  
Stationary Distributions ◽  
Birth And Death Processes ◽  
Finite State ◽  
Finite State Space ◽  
Quasi Stationary Distribution

Let qn(t) be the conditioned probability of finding a birth-and-death process in state n at time t, given that absorption into state 0 has not occurred by then. A family {q1(t), q2(t), · · ·} that is constant in time is a quasi-stationary distribution. If any exist, the quasi-stationary distributions comprise a one-parameter family related to quasi-stationary distributions of finite state-space approximations to the process.

Insensitivity in processes with zero speeds

1989 ◽  
Vol 21 (03) ◽  
pp. 612-628 ◽  
Author(s):  
P. Taylor
Keyword(s):  
Stochastic Process ◽  
Stochastic Processes ◽  
State Space ◽  
Finite State ◽  
Necessary And Sufficient ◽  
Finite State Space

Many authors have discussed the equivalence of partial balance and insensitivity in in stochastic processes. When speeds are introduced into a stochastic process there arises a difficulty in proving the necessity of partial balance for insensitivity. Previous authors have overcome this difficulty by assuming that a process has the property of instantaneous attention. This property enforces the requirement that no lifetime can be created in a state in which that lifetime has zero speed. In this paper it is shown that for processes with a finite state space it is unnecessary to make this assumption provided the notion of partial balance is slightly changed. Thus we give a criterion, analogous to partial balance, which is necessary and sufficient for insensitivity even in processes which do not possess the property of instantaneous attention. When a process does have instantaneous attention this criterion is equivalent to partial balance.

Insensitivity in processes with zero speeds

1989 ◽  
Vol 21 (3) ◽  
pp. 612-628 ◽  
Author(s):  
P. Taylor
Keyword(s):  
Stochastic Process ◽  
Stochastic Processes ◽  
State Space ◽  
Finite State ◽  
Necessary And Sufficient ◽  
Finite State Space

Many authors have discussed the equivalence of partial balance and insensitivity in in stochastic processes. When speeds are introduced into a stochastic process there arises a difficulty in proving the necessity of partial balance for insensitivity. Previous authors have overcome this difficulty by assuming that a process has the property of instantaneous attention. This property enforces the requirement that no lifetime can be created in a state in which that lifetime has zero speed.In this paper it is shown that for processes with a finite state space it is unnecessary to make this assumption provided the notion of partial balance is slightly changed. Thus we give a criterion, analogous to partial balance, which is necessary and sufficient for insensitivity even in processes which do not possess the property of instantaneous attention. When a process does have instantaneous attention this criterion is equivalent to partial balance.

Rate modulation in dams and ruin problems

1996 ◽  
Vol 33 (2) ◽  
pp. 523-535 ◽  
Author(s):  
Søren Asmussen ◽  
Offer Kella
Keyword(s):  
State Space ◽  
Release Rate ◽  
Risk Process ◽  
The State ◽  
Limiting Distribution ◽  
Process Conditions ◽  
Explicit Formulas ◽  
Rate Modulation ◽  
Finite State ◽  
Finite State Space

We consider a dam in which the release rate depends both on the state and some modulating process. Conditions for the existence of a limiting distribution are established in terms of an associated risk process. The case where the release rate is a product of the state and the modulating process is given special attention, and in particular explicit formulas are obtained for a finite state space Markov modulation.

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