Science China Mathematics

, Volume 62, Issue 4, pp 719–734 | Cite as

Exponential change of measure for general piecewise deterministic Markov processes

  • Zhaoyang Liu
  • Yuying Liu
  • Guoxin LiuEmail author


We consider a general piecewise deterministic Markov process (PDMP) X = {Xt}t⩾0 with a measure-valued generator A, for which the conditional distribution function of the inter-occurrence time is not necessarily absolutely continuous. A general form of the exponential martingales that are associated with X is given by
$$M_t^f = \frac{{f({X_t})}}{{f({X_0})}}{\left[ {S\exp (\int_{(0,t]} {\frac{{dL{{(Af)}_g}}}{{f({X_{g - }})}}} )} \right]^{ - 1}}.$$
By considering this exponential martingale to be a likelihood-ratio process, we define a new probability measure and show that the process X is still a general PDMP under the new probability measure. We additionally find the new measure-valued generator and its domain. To illustrate our results, we investigate the continuous-time compound binomial model.


exponential change of measure piecewise deterministic Markov process measure-valued generator Stieltjes exponential 


60J25 60J57 60G48 


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This work was supported by National Natural Science Foundation of China (Grant No. 11471218) and Hebei Higher School Science and Technology Research Projects (Grant No. ZD20131017).


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© Science China Press and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsCentral South UniversityChangshaChina
  2. 2.Department of Applied Mathematics and PhysicsShijiazhuang Tiedao UniversityShijiazhuangChina

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