Setting We work on a filtered probability space with finite time horizon T. The filtration...

cooloicons62

cooloicons62

Answered

2022-07-03

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 < 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?

Answer & Explanation

Elias Flores

Elias Flores

Expert

2022-07-04Added 24 answers

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 ] ]

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