Bayesian nonparametric estimation of first passage distributions in semi‐Markov processes

2019 ◽  
Vol 36 (2) ◽  
pp. 237-250
Author(s):  
Richard L. Warr ◽  
Travis B. Woodfield
Keyword(s):  
Markov Processes ◽  
Nonparametric Estimation ◽  
Bayesian Nonparametric ◽  
First Passage ◽  
Semi Markov Processes ◽  
Bayesian Nonparametric Estimation

First passage to a general threshold for a process corresponding to sampling at Poisson times

1971 ◽  
Vol 8 (03) ◽  
pp. 573-588 ◽  
Author(s):  
Barry Belkin
Keyword(s):  
Markov Processes ◽  
Classical Case ◽  
First Passage ◽  
Explicit Result ◽  
Threshold Functions ◽  
Linear Threshold Functions ◽  
Linear Threshold ◽  
Special Cases ◽  
Weiner Process ◽  
Semi Markov Processes

The problem of computing the distribution of the time of first passage to a constant threshold for special classes of stochastic processes has been the subject of considerable study. For example, Baxter and Donsker (1957) have considered the problem for processes with stationary, independent increments, Darling and Siegert (1953) for continuous Markov processes, Mehr and McFadden (1965) for Gauss-Markov processes, and Stone (1969) for semi-Markov processes. The results, however, generally express the first passage distribution in terms of transforms which can be inverted only in a relatively few special cases, such as in the classical case of the Weiner process and for certain stable and compound Poisson processes. For linear threshold functions and processes with non-negative interchangeable increments the first passage problem has been studied by Takács (1957) (an explicit result was obtained by Pyke (1959) in the special case of a Poisson process). Again in the case of a linear threshold, an explicit form for the first passage distribution was found by Slepian (1961) for the Weiner process. For the Ornstein-Uhlenbeck process and certain U-shaped thresholds the problem has recently been studied by Daniels (1969).

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