Some Bounds on the Error in Approximating Transition Probabilities in Continuous-Time Markov Processes

SIAM Review ◽  
1992 ◽  
Vol 34 (1) ◽  
pp. 110-113 ◽  
Author(s):  
John E. Angus
Keyword(s):  
Markov Processes ◽  
Continuous Time ◽  
Transition Probabilities

Continuous-time Markov processes, orthogonal polynomials and Lancaster probabilities

2020 ◽  
Vol 24 ◽  
pp. 100-112
Author(s):  
Ramsés H. Mena ◽  
Freddy Palma
Keyword(s):  
Time Dependence ◽  
Orthogonal Polynomials ◽  
Conditional Probability ◽  
Markov Processes ◽  
Continuous Time ◽  
Transition Probabilities ◽  
Spectral Expansion ◽  
Orthogonal Representation ◽  
Probability Structure

This work links the conditional probability structure of Lancaster probabilities to a construction of reversible continuous-time Markov processes. Such a task is achieved by using the spectral expansion of the corresponding transition probabilities in order to introduce a continuous time dependence in the orthogonal representation inherent to Lancaster probabilities. This relationship provides a novel methodology to build continuous-time Markov processes via Lancaster probabilities. Particular cases of well-known models are seen to fall within this approach. As a byproduct, it also unveils new identities associated to well known orthogonal polynomials.

Limit theorems for α-recurrent semi-Markov processes

1976 ◽  
Vol 8 (03) ◽  
pp. 531-547 ◽  
Author(s):  
Esa Nummelin
Keyword(s):  
Markov Processes ◽  
Continuous Time ◽  
Limit Theorems ◽  
Transition Probabilities ◽  
Renewal Processes ◽  
Markov Renewal Processes ◽  
Semi Markov Process ◽  
Limit Behaviour ◽  
Semi Markov Processes ◽  
Special Case

In this paper the limit behaviour of α-recurrent Markov renewal processes and semi-Markov processes is studied by using the recent results on the concept of α-recurrence for Markov renewal processes. Section 1 contains the preliminary results, which are needed later in the paper. In Section 2 we consider the limit behaviour of the transition probabilities Pij (t) of an α-recurrent semi-Markov process. Section 4 deals with quasi-stationarity. Our results extend the results of Cheong (1968), (1970) and of Flaspohler and Holmes (1972) to the case in which the functions to be considered are directly Riemann integrable. We also try to correct the errors we have found in these papers. As a special case from our results we consider continuous-time Markov processes in Sections 3 and 5.

Limit theorems for α-recurrent semi-Markov processes

1976 ◽  
Vol 8 (3) ◽  
pp. 531-547 ◽  
Author(s):  
Esa Nummelin
Keyword(s):  
Markov Processes ◽  
Continuous Time ◽  
Limit Theorems ◽  
Transition Probabilities ◽  
Renewal Processes ◽  
Markov Renewal Processes ◽  
Semi Markov Process ◽  
Limit Behaviour ◽  
Semi Markov Processes ◽  
Special Case

In this paper the limit behaviour of α-recurrent Markov renewal processes and semi-Markov processes is studied by using the recent results on the concept of α-recurrence for Markov renewal processes. Section 1 contains the preliminary results, which are needed later in the paper. In Section 2 we consider the limit behaviour of the transition probabilities Pij(t) of an α-recurrent semi-Markov process. Section 4 deals with quasi-stationarity. Our results extend the results of Cheong (1968), (1970) and of Flaspohler and Holmes (1972) to the case in which the functions to be considered are directly Riemann integrable. We also try to correct the errors we have found in these papers. As a special case from our results we consider continuous-time Markov processes in Sections 3 and 5.

A description of the long-term behaviour of absorbing continuous-time Markov chains using a centre manifold

1990 ◽  
Vol 22 (1) ◽  
pp. 111-128 ◽  
Author(s):  
P. K. Pollett ◽  
A. J. Roberts
Keyword(s):  
Differential Equations ◽  
Markov Chains ◽  
Invariant Manifold ◽  
Markov Processes ◽  
Continuous Time ◽  
Transition Probabilities ◽  
Centre Manifold ◽  
Continuous Time Markov Chains ◽  
Linear Differential

We use the notion of an invariant manifold to describe the long-term behaviour of absorbing continuous-time Markov processes with a denumerable infinity of states. We show that there exists an invariant manifold for the forward differential equations and we are able to describe the evolution of the state probabilities on this manifold. Our approach gives rise to a new method for calculating conditional limiting distributions, one which is also appropriate for dealing with processes whose transition probabilities satisfy a system of non-linear differential equations.

A description of the long-term behaviour of absorbing continuous-time Markov chains using a centre manifold

1990 ◽  
Vol 22 (01) ◽  
pp. 111-128 ◽  
Author(s):  
P. K. Pollett ◽  
A. J. Roberts
Keyword(s):  
Differential Equations ◽  
Markov Chains ◽  
Invariant Manifold ◽  
Markov Processes ◽  
Continuous Time ◽  
Transition Probabilities ◽  
Centre Manifold ◽  
Continuous Time Markov Chains ◽  
Linear Differential

We use the notion of an invariant manifold to describe the long-term behaviour of absorbing continuous-time Markov processes with a denumerable infinity of states. We show that there exists an invariant manifold for the forward differential equations and we are able to describe the evolution of the state probabilities on this manifold. Our approach gives rise to a new method for calculating conditional limiting distributions, one which is also appropriate for dealing with processes whose transition probabilities satisfy a system of non-linear differential equations.

The analysis of non-Markovian stochastic processes by the inclusion of supplementary variables

1955 ◽  
Vol 51 (3) ◽  
pp. 433-441 ◽  
Author(s):  
D. R. Cox
Keyword(s):  
Differential Equations ◽  
Stochastic Processes ◽  
Markov Processes ◽  
Continuous Time ◽  
Transition Probabilities ◽  
Laplace Transforms ◽  
Supplementary Variables ◽  
Discrete States

ABSTRACTCertain stochastic processes with discrete states in continuous time can be converted into Markov processes by the well-known method of including supplementary variables. It is shown that the resulting integro-differential equations simplify considerably when some distributions associated with the process have rational Laplace transforms. The results justify the formal use of complex transition probabilities. Conditions under which it is likely to be possible to obtain a solution for arbitrary distributions are examined, and the results are related briefly to other methods of investigating these processes.

scholarly journals Calibration of Transition Intensities for a Multistate Model: Application to Long-Term Care

Risks ◽  
2021 ◽  
Vol 9 (2) ◽  
pp. 37
Author(s):  
Manuel L. Esquível ◽  
Gracinda R. Guerreiro ◽  
Matilde C. Oliveira ◽  
Pedro Corte Real
Keyword(s):  
Markov Chain ◽  
Continuous Time ◽  
Long Term Care ◽  
Transition Probabilities ◽  
Markov Chain Model ◽  
Chain Model ◽  
Chain Process ◽  
Term Care ◽  
National Network

We consider a non-homogeneous continuous time Markov chain model for Long-Term Care with five states: the autonomous state, three dependent states of light, moderate and severe dependence levels and the death state. For a general approach, we allow for non null intensities for all the returns from higher dependence levels to all lesser dependencies in the multi-state model. Using data from the 2015 Portuguese National Network of Continuous Care database, as the main research contribution of this paper, we propose a method to calibrate transition intensities with the one step transition probabilities estimated from data. This allows us to use non-homogeneous continuous time Markov chains for modeling Long-Term Care. We solve numerically the Kolmogorov forward differential equations in order to obtain continuous time transition probabilities. We assess the quality of the calibration using the Portuguese life expectancies. Based on reasonable monthly costs for each dependence state we compute, by Monte Carlo simulation, trajectories of the Markov chain process and derive relevant information for model validation and premium calculation.

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