Stochastic indistinguishability

2020 Mathematics Subject Classification: Primary: 60Gxx [MSN][ZBL]

A property of two random processes $ X = ( X _ {t} ( \omega )) _ {t \geq 0 } $ and $ Y = ( Y _ {t} ( \omega )) _ {t \geq 0 } $ which states that the random set

$$ \{ X \neq Y \} = \ \{ {( \omega , t) } : {X _ {t} ( \omega ) \neq Y _ {t} ( \omega ) } \} $$

can be disregarded, i.e. that the probability of the set $ \{ \omega : {\exists t \geq 0 : ( \omega , t) \in \{ X \neq Y \} } \} $ is equal to zero. If $ X $ and $ Y $ are stochastically indistinguishable, then $ X _ {t} = Y _ {t} $ for all $ t \geq 0 $, i.e. $ X $ and $ Y $ are stochastically equivalent (cf. Stochastic equivalence). The opposite, generally speaking, is not true, but for processes that are continuous from the right (left), stochastic indistinguishability follows from stochastic equivalence.

References

[D]	C. Dellacherie, "Capacités et processus stochastiques" , Springer (1972) MR0448504 Zbl 0246.60032

References

[DM]	C. Dellacherie, P.A. Meyer, "Probabilities and potential" , A , North-Holland (1978) (Translated from French) MR0521810 Zbl 0494.60001