In the mathematical theory of probability, a Borel right process, named after Émile Borel, is a particular kind of continuous-time random process.

Let be a locally compact, separable, metric space. We denote by the Borel subsets of . Let be the space of right continuous maps from to that have left limits in , and for each , denote by the coordinate map at ; for each , is the value of at . We denote the universal completion of by . For each , let

and then, let

For each Borel measurable function on , define, for each ,

Since and the mapping given by is right continuous, we see that for any uniformly continuous function , we have the mapping given by is right continuous.

Therefore, together with the monotone class theorem, for any universally measurable function , the mapping given by , is jointly measurable, that is, measurable, and subsequently, the mapping is also -measurable for all finite measures on and on . Here, is the completion of with respect to the product measure . Thus, for any bounded universally measurable function on , the mapping is Lebeague measurable, and hence, for each , one can define

There is enough joint measurability to check that is a Markov resolvent on , which uniquely associated with the Markovian semigroup . Consequently, one may apply Fubini's theorem to see that

The following are the defining properties of Borel right processes:[1]

  • Hypothesis Droite 1:
For each probability measure on , there exists a probability measure on such that is a Markov process with initial measure and transition semigroup .
  • Hypothesis Droite 2:
Let be -excessive for the resolvent on . Then, for each probability measure on , a mapping given by is almost surely right continuous on .

Notes

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  1. ^ Sharpe 1988, Sect. 20

References

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  • Sharpe, Michael (1988), General Theory of Markov Processes, ISBN 0126390606