A martingale characterization of the Wiener process

1994 ◽  
Vol 19 (2) ◽  
pp. 167
Author(s):  
Jacek Wesołowski
Keyword(s):  
Wiener Process ◽  
Martingale Characterization

A martingale characterization of mixed Poisson processes

1987 ◽  
Vol 24 (1) ◽  
pp. 246-251 ◽  
Author(s):  
Dietmar Pfeifer ◽  
Ursula Heller
Keyword(s):  
Poisson Process ◽  
Random Variable ◽  
Poisson Processes ◽  
Birth Process ◽  
Mixed Poisson Process ◽  
Martingale Characterization

It is shown that an elementary pure birth process is a mixed Poisson process iff the sequence of post-jump intensities forms a martingale with respect to the σ -fields generated by the jump times of the process. In this case, the post-jump intensities converge almost surely to the mixing random variable of the process.

scholarly journals A martingale characterization of Pólya-Lundberg processes

2006 ◽  
Vol 43 (03) ◽  
pp. 741-754 ◽  
Author(s):  
Birgit Niese
Keyword(s):  
Poisson Process ◽  
Exponential Family ◽  
Exponential Families ◽  
Counting Processes ◽  
Mixed Poisson Process ◽  
Martingale Characterization

We study exponential families within the class of counting processes and show that a mixed Poisson process belongs to an exponential family if and only if it is either a Poisson process or has a gamma structure distribution. This property can be expressed via exponential martingales.

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