# Stochastic indistinguishability

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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