# Brownian motion, modifications vs indistinguishablity

In Protter's book Stochastic Integration and Differential Equations and in uncountable other sources, they mention the continuous sample paths of the brownian motion. That is:

It holds that $$t\mapsto B_t(\omega)$$ is a continous curve.

However, in the literature I have, they only prove existence of a continuous modification Y of a Brownian motion X. By definition, that is

$$P(X_t\neq Y_t)=0 \quad \forall t\in[0,\infty).$$ As Protter himself mention on page 4, we cannot say anything about the set $$\bigcup_{t\geq 0}(X_t\neq Y_t),$$ when we discuss modifications, since this is an uncountable union. But is it really true, that the version of the brownian motion which are continuous, and that we choose to use for further investigation, might differ significantly from the original brownian motion X? (More precisely, might the set where they differ not be measurable, or have positive probability?)

• Yes I believe this is correct. Jul 15, 2016 at 18:12

As stated, the question isn't well-posed. Brownian motion is a concept of a stochastic process with a set of properties which only uniquely determine this process in a suitable sense.

If $(\Omega,\mathcal A,\operatorname P)$ is a probability space and $(\mathcal F_t)_{t\ge0}$ is a filtration of $\mathcal A$, $B:\Omega\times[0,\infty)\to\mathbb R$ is called Brownian motion on $(\Omega,\mathcal A,\operatorname P)$ with respect to $\mathcal F$ iff

1. $B_0=0$
2. $B$ is continuous
3. $B_t-B_s\sim\mathcal N_{0,\:t-s}$ and $\mathcal F_s$ are independent, for all $t>s\ge0$

There are various ways to construct and hence prove the existence of such an object. Most probably, in your literature, the author(s) constructed an object with (1) and (3) and then mentioned that this object has a continuous modification and hence is a Brownian motion in the sense defined above.

Now, you can replace (1) and (2) by

1. $B_0=0$ almost surely
2. $B$ is continuous almost surely

The point is, that it doesn't matter, cause two almost surely continuous processes which are modifications of each other are already indistinguishable. In other words, $B$ is uniquely determined up to $\operatorname P$-indistinguishability. Hence, you can always assume that these properties hold surely.