cooloicons62

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 $P\left[{X}_{T}\ge 0\right]=1$.
Question I would like to obtain that $X$ (up to indistinguishability) satisfies property (A) for each $\omega$.
My attempt Define a process $Y$ to be equal to 0 on the event $\left\{{X}_{T}<0\right\}$ and define it to be equal to $X$ on $\left\{{X}_{T}\ge 0\right\}$. Then ${Y}_{T}\ge 0$. Is it then correct that $X,Y$ are indistinguishable, i.e. $P\left[{X}_{t}={Y}_{t},\mathrm{\forall }t\in \left[0,T\right]\right]=1$? Is this approach possible whenever the event where the stochastic process does not satisfy the desired propert has probability 0?

Elias Flores

Expert

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 $\omega$ 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:
$=\left\{{X}_{T}={Y}_{T}\right\}\cup \left\{{X}_{T}\ne {Y}_{T}\right\}$={XT=YT}∪{XT≠YT}
So as $P\left[{X}_{T}={Y}_{T}\right]=1=P\left[{X}_{t}={Y}_{t},\mathrm{\forall }t\in \left[0,T\right]\right]$

Do you have a similar question?