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.

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