This is taken from example 1.4 in Brownian Motion and Stochastic calculus, by Karatzas and Shreve. There's obviously something I've misunderstood, but here goes:

Assume that $T$ is a positive random variable with a continuous distribution. Let $(X_t)_{t\ge 0},(Y_t)_{t\ge 0}$ be two stochastic processes where $X_t = 0\,\, \forall t\ge 0$ and

$$Y_t = \begin{cases} 0, & t \neq T, \\ 1, & t = T. \end{cases}$$

Then $X$ and $Y$ are modifications of each other since

$$\forall t\ge 0\colon \, \mathbb{P}(X_t = Y_t) = \mathbb{P}(t \neq T) = 1.$$

However, $\mathbb{P}(\forall t\ge 0\colon X_t = Y_t) = 0$, so the processes aren't indistinguishable.

My problem: There is a result that says if $X$ and $Y$ are modifications and have $\mathbb{P}$-a.s. right-continuous sample paths, then $X$ and $Y$ are indistinguishable.

Here, $t \mapsto X_t$ is constant, so it must be continuous, and $t \mapsto Y_t$ is discontinuous only on a null set, so it must be $\mathbb{P}$-a.s. continuous.

Hence $X,Y$ should be indistinguishable, no?


1 Answer 1


It is true that the paths of $Y$ all have only one point of discontinuity (namely the value of $T$), but that still means that all paths of $Y$ are not (right-)continuous, so in fact $Y$ is right-continuous with probability $0$, which is why the result you cited does not apply here.

You were confusing null sets $B\subset \mathbb{R}$ with respect to Lebesgue measure and null sets $N\subset \Omega$ with respect to the probability measure $\mathbb{P}$ (where $\mathbb{P}$ is defined on some measurable space $(\Omega,\mathcal{A})$).


You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .