Redudant condition for Wiener process: continuity?

Question

In my book, a Wiener process is defined as a process which has, amongst others, the conditions $W_t - W_s \sim \mathcal{N}(0,t - s), t > s,$ and that the process has a "continuous trajectory".

First of all, what is meant by a "continuous trajectory"? Is it just that $|W_t - W_s| < \epsilon$ if $|t - s|< \delta$? If so, is this condition not redundant given the Gaussian distribution of the increments, which, when $|t-s|$ is small, approaches the constant $0$?

Answer 1

First of all, what is meant by "continuous trajectory"?

The trajectory (also: sample path) of a Wiener process is the mapping $t \mapsto W_t(\omega)$ for fixed $\omega \in \Omega$. A Wiener process has (almost surely) continuous sample paths, i.e. the mappings $t \mapsto W_t(\omega)$ are continuous for (almost) all $\omega \in \Omega$.

[...] If so, is this condition not redunant given the Gaussian distribution of the increments, which, when $|t-s|$ is small, aprooaches the constant $0$.

No, it's not. Using this distributional property, it is possible to show that $(W_t)_{t \geq 0}$ is continuous in probability (in the sense that $W_s \to W_t$ in probability as $s \to t$); this is weaker than assuming (pointwise) continuity of the trajectories. However, it is possible to show the following statement:

Let $(W_t)_{t \geq 0}$ be a stochastic process which satisfies $W_t-W_s \sim N(0,t-s)$ for all $s \leq t$, then $(W_t)_{t \geq 0}$ has a modification with exclusively continuous trajectories.

This follows e.g. from a well-known result by Kolmogorov and Chentsov.