Setting We work on a filtered probability space with finite time horizon T. The filtration is assumed to be complete. Let X be a stochastic process that satisfies a property (A) a.s. For example, if property (A) is being nonnegative at the time T, then X satisfies P [ X T ≥ 0 ] = 1.Question I would like to obtain that X (up to indistinguishability) satisfies property (A) for each ω.My attempt Define a process Y to be equal to 0 on the event { X T &lt; 0 } and define it to be equal to X on { X T ≥ 0 }. Then Y T ≥ 0. Is it then correct that X , Y are indistinguishable, i.e. P [ X t = Y t , ∀ t ∈ [ 0 , T ] ] = 1? Is this approach possible whenever the event where the stochastic process does not satisfy the desired propert has probability 0?

Question

Setting We work on a filtered probability space with finite time horizon   T. The filtration is assumed to be complete. Let   X be a stochastic process that satisfies a property (A) a.s. For example, if property (A) is being nonnegative at the time   T, then   X satisfies   P  [      X    T    ≥  0  ]  =  1.Question I would like to obtain that   X (up to indistinguishability) satisfies property (A) for each   ω.My attempt Define a process   Y to be equal to 0 on the event   {      X    T    &amp;lt;  0  } and define it to be equal to   X on   {      X    T    ≥  0  }. Then       Y    T    ≥  0. Is it then correct that   X  ,  Y are indistinguishable, i.e.   P  [      X    t    =      Y    t    ,  ∀  t  ∈  [  0  ,  T  ]  ]  =  1? Is this approach possible whenever the event where the stochastic process does not satisfy the desired propert has probability 0?

Elias Flores · Accepted Answer

Well if you modify   X only at time   T over an event of 0 mass, yes all that happens strictly before   T is equal to   X for every   ω and only for the path which are negative at time   T you have to modify your process so yes   X and   Y are indistinguishable. Formally:  =  {      X    T    =      Y    T    }  ∪  {      X    T    ≠      Y    T    }={XT=YT}∪{XT≠YT}So as   P  [      X    T    =      Y    T    ]  =  1  =  P  [      X    t    =      Y    t    ,  ∀  t  ∈  [  0  ,  T  ]  ]