A use of complex probabilities in the theory of stochastic processes

1955 ◽  
Vol 51 (2) ◽  
pp. 313-319 ◽  
Author(s):  
D. R. Cox
Keyword(s):  
Stochastic Processes ◽  
Exponential Distribution ◽  
Continuous Time ◽  
Transition Probabilities ◽  
Laplace Transforms ◽  
Exponential Distributions ◽  
Discrete States

ABSTRACTThe exponential distribution is very important in the theory of stochastic processes with discrete states in continuous time. A. K. Erlang suggested a method of extending to other distributions methods that apply in the first instance only to exponential distributions. His idea is generalized to cover all distributions with rational Laplace transforms; this involves the formal use of complex transition probabilities. Properties of the method are considered.

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.

On an approximation made when analysing stochastic processes

1976 ◽  
Vol 13 (4) ◽  
pp. 672-683 ◽  
Author(s):  
Byron J. T. Morgan ◽  
John P. Hinde
Keyword(s):  
Markov Process ◽  
Stochastic Processes ◽  
Continuous Time ◽  
Discrete States

We investigate the effect of a particular mode of approximation by means of four examples of its use; in each case the model approximated is a Markov process with discrete states in continuous time.

On an approximation made when analysing stochastic processes

1976 ◽  
Vol 13 (04) ◽  
pp. 672-683 ◽  
Author(s):  
Byron J. T. Morgan ◽  
John P. Hinde
Keyword(s):  
Markov Process ◽  
Stochastic Processes ◽  
Continuous Time ◽  
Discrete States

We investigate the effect of a particular mode of approximation by means of four examples of its use; in each case the model approximated is a Markov process with discrete states in continuous time.

Ehrenfest urn models

1965 ◽  
Vol 2 (02) ◽  
pp. 352-376 ◽  
Author(s):  
Samuel Karlin ◽  
James McGregor
Keyword(s):  
Exponential Distribution ◽  
Continuous Time ◽  
Random Variables ◽  
Independent Random Variables ◽  
Urn Models ◽  
Time Intervals ◽  
Negative Exponential Distribution ◽  
Negative Exponential ◽  
Random Times

In the Ehrenfest model with continuous time one considers two urns and N balls distributed in the urns. The system is said to be in stateiif there areiballs in urn I, N −iballs in urn II. Events occur at random times and the time intervals T between successive events are independent random variables all with the same negative exponential distributionWhen an event occurs a ball is chosen at random (each of theNballs has probability 1/Nto be chosen), removed from its urn, and then placed in urn I with probabilityp, in urn II with probabilityq= 1 −p, (0 <p< 1).

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.

Univariate and bivariate extensions of the generalized exponential distributions

2021 ◽  
Vol 71 (6) ◽  
pp. 1581-1598
Author(s):  
Vahid Nekoukhou ◽  
Ashkan Khalifeh ◽  
Hamid Bidram
Keyword(s):  
Em Algorithm ◽  
Exponential Distribution ◽  
Bivariate Distribution ◽  
Generalized Exponential Distribution ◽  
Exponential Distributions ◽  
Unknown Parameters ◽  
Data Set ◽  
New Class ◽  
Special Cases ◽  
Univariate Distribution

Abstract The main aim of this paper is to introduce a new class of continuous generalized exponential distributions, both for the univariate and bivariate cases. This new class of distributions contains some newly developed distributions as special cases, such as the univariate and also bivariate geometric generalized exponential distribution and the exponential-discrete generalized exponential distribution. Several properties of the proposed univariate and bivariate distributions, and their physical interpretations, are investigated. The univariate distribution has four parameters, whereas the bivariate distribution has five parameters. We propose to use an EM algorithm to estimate the unknown parameters. According to extensive simulation studies, we see that the effectiveness of the proposed algorithm, and the performance is quite satisfactory. A bivariate data set is analyzed and it is observed that the proposed models and the EM algorithm work quite well in practice.

scholarly journals An alternative characterization for matrix exponential distributions

2009 ◽  
Vol 41 (04) ◽  
pp. 1005-1022
Author(s):  
Mark Fackrell
Keyword(s):  
Continuous Function ◽  
Exponential Distribution ◽  
Matrix Exponential ◽  
Necessary Condition ◽  
Stieltjes Transform ◽  
Real Numbers ◽  
Exponential Distributions ◽  
Alternative Characterization

A necessary condition for a rational Laplace–Stieltjes transform to correspond to a matrix exponential distribution is that the pole of maximal real part is real and negative. Given a rational Laplace–Stieltjes transform with such a pole, we present a method to determine whether or not the numerator polynomial admits a transform that corresponds to a matrix exponential distribution. The method relies on the minimization of a continuous function of one variable over the nonnegative real numbers. Using this approach, we give an alternative characterization for all matrix exponential distributions of order three.

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