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A stochastic process $\{W(t): t \in T\}$ is a Wiener process if it satisfies the following properties:

1) $W(0)=0$ with probability $1$

2) It has stationary and independent increments.

3) For every $t>0, W(t)\text{ ~ }N(0,t)$

4) It has continuous paths.

My question is , what does "it has continuous paths" mean? How can I define that?

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    $\begingroup$ For almost every $\omega$ in $\Omega$, the function $t\mapsto W(t)(\omega)$ is continuous. $\endgroup$ – Did Sep 27 '15 at 16:55
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Well in general there are multiple ways to introduce a Wiener process and the answer will depend on that. One construction is via Kolmogorov's extension theorem. With help of the latter you can construct a Brownian motion, which essentially is a measurable function:

$$ W:(\Omega, \mathcal{F}, \mathbb P) \to \mathbb R^{[0,1]} $$

In other words note that $W(\omega)$ for $\omega \in \Omega$ is an element of $\mathbb R^{[0,1]}$, i.e. a function $W(\omega):[0,1] \to \mathbb R, t \mapsto W(\omega)(t)$. Since your random elements are functions, you can of course say whether they are continuous or not, i.e. you can check whether $W(\omega)$ is continuous or not.

Now you combine this with the fact that your functions are actually random to make statements such as "for almost every $\omega \in \Omega$ the function $W(\omega)$ is continuous". This is what we mean by "it has continuous paths" (usually it is stated "it almost surely has continued paths"). (In other words we call a "path" a random function $W(\omega)$).

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